Lagrangian and Hamiltonian geometries. Applications … · arXiv:1203.4101v1 [math.DG] 19 Mar 2012 Radu MIRON Lagrangian and Hamiltonian geometries. Applications to Analytical Mechanics

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Radu MIRON

Lagrangian and Hamiltoniangeometries. Applications toAnalytical Mechanics

http://arxiv.org/abs/1203.4101v1

Preface

The aim of the present monograph is twofold: 1 to provide a Com-pendium of Lagrangian and Hamiltonian geometries and 2 to in-troduce and investigate new analytical Mechanics: Finslerian, La-grangian and Hamiltonian.

One knows (R. Abraham, J. Klein, R. Miron et al.) that the ge-ometrical theory of nonconservative mechanical systems can not berigourously constructed without the use of the geometry of the tan-gent bundle of the configuration space.

The solution of this problem is based on the Lagrangian and Hamil-tonian geometries. In fact, the construction of these geometries relieson the mechanical principles and on the notion of Legendre transfor-mation.

The whole edifice has as support the sequence of inclusions:

Rn ⊂ Fn ⊂ Ln ⊂ GLn

formed by Riemannian, Finslerian, Lagrangian, and generalized La-grangian spaces. TheL−duality transforms this sequence into a sim-ilar one formed by Hamiltonian spaces.

Of course, these sequences suggest the introduction of the corre-spondent Mechanics: Riemannian, Finslerian, Lagrangian,Hamilto-nian etc.

The fundamental equations (or evolution equations) of these Me-chanics are derived from the variational calculus applied to the inte-gral of action and these can be studied by using the methods ofLa-grangian or Hamiltonian geometries.

More general, the notions of higher order Lagrange or Hamiltonspaces have been introduced by the present author [161], [162], [163]and developed is realized by means of two sequences of inclusionssimilarly with those of the geometry of order 1. The problemsraisedby the geometrical theory of Lagrange and Hamilton spaces oforderk ≥ 1 have been investigated by Ch. Ehresmann, W. M. Tulczyjew,A. Kawaguchi, K. Yano, M. Crampin, Manuel de Leon, R. Miron,

v

vi Preface

M. Anastasiei, I. Bucataru et al. [175]. The applications lead to thenotions of Lagrangian or Hamiltonian Analytical Mechanicsof orderk.

For short, in this monograph we aim to solve some difficult prob-lems:

- The problem of geometrization of classical non conservative me-chanical systems;

- The foundations of geometrical theory of new mechanics: Finsle-rian, Lagrangian and Hamiltonian;

- To determine the evolution equations of the classical mechanicalsystems for whose external forces depend on the higher orderacceler-ations.

This monograph is based on the terminology and important re-sults taken from the books: Abraham, R., Marsden, J.,Foundation ofMechanics, Benjamin, New York, 1978; Arnold, V.I.,MathematicalMethods in Classical Mechanics, Graduate Texts in Math, SpringerVerlag, 1989; Asanov, G.S.,Finsler Geometry Relativity and Gaugetheories, D. Reidel Publ. Co, Dordrecht, 1985; Bao, D., Chern, S.S.,Shen, Z.,An Introduction to Riemann-Finsler Geometry, SpringerVerlag, Grad. Text in Math, 2000; Bucataru, I., Miron. R.,Finsler-Lagrange geometry. Applications to dynamical systems, Ed. AcademieiRomane, 2007; Miron, R., Anastasiei, M.,The geometry of Lagrangespaces. Theory and applications to Relativity, Kluwer Acad. Publ.FTPH no. 59, 1994; Miron, R.,The Geometry of Higher-Order La-grange Spaces. Applications to Mechanics and Physics, Kluwer Acad.Publ. FTPH no. 82, 1997; Miron, R.,The Geometry of Higher-OrderFinsler Spaces, Hadronic Press Inc., SUA, 1998; Miron, R., Hrimiuc,D., Shimada, H., Sabau, S.,The geometry of Hamilton and LagrangeSpaces, Kluwer Acad. Publ., FTPH no. 118, 2001.

This book has been received a great support from Prof. OvidiuCarja, the dean of the Faculty of Mathematics from “Alexandru IoanCuza” University of Iasi.

My colleagues Professors M. Anastasiei, I. Bucataru, I. Mihai, M.Postolache, K. Stepanovici made important remarks and suggestionsand Mrs. Carmen Savin prepared an excellent print-form of the hand-written text. Many thanks to all of them.

Iasi, June 2011 Radu Miron

Preface vii

The purpose of the book is a short presentation of the geometricaltheory of Lagrange and Hamilton spaces of order 1 or of orderk≥ 1,as well as the definition and investigation of some new AnalyticalMechanics of Lagrangian and Hamiltonian type of orderk≥ 1.

In the last thirty five years, geometers, mechanicians and physicistsfrom all over the world worked in the field of Lagrange or Hamiltongeometries and their applications. We mention only some importantnames: P.L. Antonelli [21], M. Anastasiei [10], [11]. G. S. Asanov[27], A. Bejancu [38], I. Bucataru [47], M. Crampin [64], R.S. In-garden [114], S. Ikeda [116], M. de Leon [138], M. Matsumoto [144],R. Miron [164], [165], [166], H. Rund [218], H. Shimada [223], P.Stavrinos [237], L. Tamassy [247] and S. Vacaru [251].

The book is divided in three parts: I. Lagrange and Hamiltonspaces; II. Lagrange and Hamilton spaces of higher order; III. Analy-tical Mechanics of Lagrangian and Hamiltonian mechanical systems.

The part I starts with the geometry of tangent bundle(TM,π ,M)of a differentiable, real,n−dimensional manifoldM. The main geo-metrical objects onTM, as Liouville vector fieldC, tangent structureJ, semisprayS, nonlinear connectionN determined byS, N−metricalstructureD are pointed out. It is continued with the notion of La-grange space, defined by a pairLn = (M,L(x,y)) with L : TM →R as

a regular Lagrangian andgi j =12

∂i ∂ jL as fundamental tensor field. Of

course,L(x,y) is a regular Lagrangian if the Hessian matrix‖gi j (x,y)‖is nonsingular. In the definition of Lagrange spaceLn we assume thatthe fundamental tensorgi j (x,y) has a constant signature. The knownLagrangian from Electrodynamics assures the existence of Lagrangespaces.

The variational problem associated to the integral of action

I(c) =

1∫

0

L(x, x)dt

allows us to determine the Euler–Lagrange equations, conservationlaw of energy,EL, as well as the canonical semisprayS of Ln. ButS determines the canonical nonlinear connectionN and the metricalN−linear connectionD, given by the generalized Christoffel symbols.The structure equations ofD are derived. This theory is applied to thestudy of the electromagnetic and gravitational fields of thespaceLn.An almost Kahlerian model is constructed. This theory suggests todefine the notion of generalized Lagrange spaceGLn = (M,g(x,y)),whereg(x,y) is a metric tensor on the manifoldTM. The spaceGLn

is not reducible to a spaceLn.A particular case of Lagrange spaceLn leads to the known con-

cept of Finsler spaceFn = (M,F(x,y)). It follows that the geometry

viii Preface

of Finsler spaceFn can be constructed only by means of AnalyticalMechanics principles.

Since a Riemann spaceRn = (M ·g(x)) is a particular Finsler spaceFn = (M,F(x,y)) we get the following sequence of inclusions:

(I) Rn ⊂ Fn ⊂ Ln ⊂ GLn.

The Lagrangian geometry is the geometrical study of the sequence (I).The geometrical theory of the Hamilton spaces can be constructed

step by step following the theory of Lagrange spaces. The legiti-macy of this theory is due toL−duality (Legendre duality) be-tween a Lagrange spaceLn = (M,L(x,y)) and a Hamilton spaceHn = (M,H(x, p)).

Therefore, we begin with the geometrical theory of cotangent bun-dle (T∗M,π∗,M), continue with the notion of Hamilton spaceHn(M,H(x, p)), whereH : T∗M → R is a regular Hamiltonian, with the vari-ational problem

I(c) =∫ 1

0

[pi(t)

dxi

dt−

12

L(x(t), p(t))

]dt,

from which the Hamilton–Jacobi equations are derived, Hamiltonianvector field ofHn, nonlinear connectionN, N∗−linear connection etc.The Legendre transformationL : Ln → Hn transforms the fundamen-tal geometrical object fields onLn into the fundamental geometricalobject fields onHn. The restriction ofL to the Finsler spacesFnhas as image a new class of spacesC n = (M,K(x, p)) called the Car-tan spaces. They have the same beauty, symmetry and importance asthe Finsler spaces. A pairGHn = (M,g∗(x, p)), whereg∗ is a metrictensor onT∗M is named a generalized Hamilton space. Remarkingthat R∗n = (M,g∗(x)), with g∗i j (x) the contravariant Cartan space,we obtain a dual sequence of the sequence (I):

(II) R∗n ⊂ C n ⊂ Hn ⊂ GHn.

The Hamiltonian geometry is geometrical study of the sequence II.The Lagrangian and Hamiltonian geometries are useful for applica-

tions in: Variational calculus, Mechanics, Physics, Biology etc.The part II of the book is devoted to the notions of Lagrange and

Hamilton spaces of higher order. We study the geometrical theory ofthe total space ofk−tangent bundleTkM, k≥ 1, generalizing, step bystep the theory from casek = 1. So, we introduce the Liouville vec-tor fields, study the variational problem for a given integral of actionon TkM, continue with the notions ofk−semispray, nonlinear con-

Preface ix

nection, the prolongation toTkM of the Riemannian structure definedon the base manifoldM. The notion ofN−metrical connections ispointed out, too.

It follows the notion of Lagrange space of orderk≥ 1. It is definedas a pairL(k)n = (M,L(x,y(1),y(2), ...,y(k))), whereL is a Lagrangiandepending on the (material point)x = (xi) and on the accelerations

y(k)i=

1k!

dkxi

dt, k = 1,2, ...,k; i = 1,2, ...,n, n = dimM. Some exam-

ples prove the existence of the spacesL(k)n. The canonical nonlinearconnectionN and metricalN−linear connection are pointed out. TheRiemannian almost contact model forL(k)n as well as the GeneralizedLagrange space of orderk end this theory.

The methods used in the construction of the Lagrangian geometryof higher-order are the natural extensions of those used in the theoryof Lagrangian geometries of orderk= 1.

Next chapter treats the geometry of Finsler spaces of orderk ≥ 1,F(k)1 = (M,F(x,y(1), ...,y(k))), F : TkM → Rbeing positivek−homo-geneous on the fibres ofTkM and fundamental tensor has a constantsignature. Finally, one obtains the sequences

(III ) R(k)n ⊂ F(k)n ⊂ L(k)1 ⊂ GL(k)n.

The Lagrangian geometry of orderk≥ 1 is the geometrical theory ofthe sequences of inclusions (III).

The notion of Finsler spaces of orderk, introduced by the presentauthor, was investigated in the bookThe Geometry of Higher-OrderFinsler Spaces, Hadronic Press, 1998. Here it is a natural extension to

the manifoldTkM of the theory of Finsler spaces given in Section 3.It was developed by H. Shimada and S. Sabau [223].

The previous theory is continued with the geometry ofk cotangentbundleT∗kM, defined as the fibered product:Tk−1MXMT∗M, with thenotion of Hamilton space of orderk ≥ 1 and with the particular caseof Cartan spaces of orderk. An extension toL(k)n andH(k)n of theLegendre transformation is pointed out. The Hamilton–Jacobi equa-tions of H(k)n, which are fundamental equations of these spaces, arepresented, too.

For the Hamilton spaces of orderk, H(k)n = (M, H(x,y)(k), ...,y(k−1), p) a similar sequence of inclusions with (III) is introduced.These considerations allow to clearly define what means the Hamil-tonian geometry of orderk and what is the utility of this theory inapplications.

Part III of this book is devoted to applications in Analytical Me-chanics. One studies the geometrical theory of scleronomicnoncon-

x Preface

servative classical mechanical systemsΣR = (M,T,Fe), it is intro-duced and investigated the notion of Finslerian mechanicalsystemsΣF = (M,F,Fe) and is defined the concept of Lagrangian mechan-ical systemΣL = (M,L,Fe). In all these theoriesM is the con-figuration space,T is the kinetic energy of a Riemannian spaceRn = (M,g), F(x,y) is the fundamental function of a Finsler spaceFn = (M,F(x,y)), L(x,y) is a regular Lagrangian andFe(x,y) are theexternal forces which depend on the material pointsx ∈ M and their

velocitiesyi = xi =dxi

dt.

In order to study these Mechanics we apply the methods from theLagrangian geometries. The contents of these geometrical theory ofmechanical systemsΣR , ΣF andΣL is based on the geometrical theoryof the velocity spaceTM. The base of these investigations are theLagrange equations. They determine a canonical semispray,which isfundamental for all constructions. For every case ofΣR , ΣF andΣLthe law of conservation of energy is pointed out. We end with thecorresponding almost Hermitian model on the velocity spaceTM.

The dual theory via Legendre transformation, leads to the geometri-cal study of the Hamiltonian mechanical systemsΣ∗

R= (M,T∗,Fe∗),

Σ∗C= (M,K,Fe∗) andΣ∗

H = (M,H,Fe∗) whereT∗ is energy,K(x, p)is the fundamental function of a given Cartan space andH(x, p) is aregular Hamiltonian on the cotangent bundleT∗M. The fundamentalequations in these Hamiltonian mechanical systems are the Hamiltonequations

dxi

dt=−

∂H

∂ pi;

dpi

dt=

∂H

∂xi +12

Fi ,

(H =

12

H

),

Fi(x, p) being the covariant components of external forces. The usedmethods are those given by the sequence of inclusionsR∗⊂C n⊂Hn ⊂ GHn.

More general, the notions of Lagrangian and Hamiltonian mechan-ical systems of orderk ≥ 1 are introduced and studied. Therefore, tothe sequences of inclusions III correspond the analytical mechanicsof orderk of the Riemannian, Finslerian, Lagrangian type. All thesecases are the direct generalizations from casek= 1. This is the reasonwhy we present here shortly the principal results. Especially, we paidattention to the following question:

Considering a Riemannian(particularly Euclidian) mechanical

systemΣR = (M,T,Fe), with T =12

gi j (x)xi x j as energy, but with the

external forces Fe depending on material point(xi) and on higher

Preface xi

order accelerations

(dxi

dt,

d2xi

dt2, ....

dkxi

dtk

). What are the evolution

equations of the systemΣR?

Clearly, the classical Lagrange equations are not valid. Inthe last partof this book we present the solution of this problem.

Finally, what is new in the present book?

1A solution of the problem of geometrization of the classicalnon-conservative mechanical systems, whose external forces depend onvelocities, based on the differential geometry of velocityspace.

2The introduction of the notion of Finslerian mechanical system.3The definition of Cartan mechanical system.4The study of theory of Lagrangian and Hamiltonian mechanical

systems by means of the geometry of tangent and cotangent bun-dles.

5The geometrization of the higher order Lagrangian and Hamilto-nian mechanical systems.

6The determination of the fundamental equations of the Riemannianmechanical systems whose external forces depend on the higherorder accelerations.

Part ILagrange and Hamilton Spaces

The purpose of the part I is a short presentation of the geometrical the-ory of Lagrange space and of Hamilton spaces. These spaces are basicfor applications to Mechanics, Physics etc. and have been introducedby the present author in 1980 and 1987, respectively.

Therefore we present here the general framework of LagrangeandHamilton geometries based on the books by R. Miron [161], [162],[163], R. Miron and M. Anastasiei [164], I. Bucataru and R. Miron[49] and R. Miron, D. Hrimiuc, H. Shimada and S. Sabau [174].Also,we use the papers R. Miron, M. Anastasiei and I. Bucataru,The ge-ometry of Lagrange spaces, Handbook of Finsler Geometry, P. L. An-tonelli ed., Kluwer Academic; R. Miron,Compendium on the Geom-etry of Lagrange spaces, Handbook of Differential Geometry, vol. II,pp 438–512, Edited by F. J. E. Dillen and L. C. A. Verstraelen,2006.

The geometry of Lagrange space starts here with the study of geo-metrical theory of tangent bundle(TM,π ,M). It is continued with thenotion of Lagrange spaceLn = (M,L(x,y)) and with an important par-ticular case the Finsler spaceFn = (M,F). The Lagrangian geometryis the study of the sequence of inclusions (I) from the Preface.

The geometry of Hamilton spaces follow the same pattern: thege-ometry of cotangent bundle(T∗M,π∗,M), it continues with the no-tion of Hamilton spaceHn = (M,H(x, p)), with the concept of CartanspaceC = (M,K(x, p)) and ends with the sequence of inclusions (II)from Introduction.

The relation between the previous sequences are given by means ofthe Legendre transformation.

In the following, we assume that all the geometrical object fieldsand mappings areC∞−differentiable and we express this by words“differentiable” or “smooth”.

Chapter 1The Geometry of tangent manifold

The total spaceTM of tangent bundle(TM,π ,M) carries some naturalobject fields as Liouville vector fieldC, tangent structureJ and verti-cal distributionV. An important object field is the semispraySdefinedas a vector fieldS on the manifoldTM with the propertyJ(S) = C.One can develop a consistent geometry of the pair(TM,S).

1.1 The manifold TM

The differentiable structure onTM is induced by that of the base man-ifold M so that the natural projectionπ : TM → M is a differentiablesubmersion and the triple(TM,π,M) is a differentiable vector bun-dle. Assuming thatM is a real,n−dimensional differentiable mani-fold and(U,ϕ = (xi)) is a local chart at a pointx∈ M, then any curveσ : I → M, (Imσ ⊂ U ) that passes throughx at t = 0 is analyticallyrepresented byxi = xi(t), t ∈ I , ϕ(x) = (xi(0)), (i, j, ...= 1, ...,n). Thetangent vector[σ ]x is determined by the real numbers

xi = xi(0), yi =dxi

dt(0).

Then the pair(π−1(U),Φ), with Φ([σ ]x) = (xi ,yi) ∈ R2n, ∀[σ ]x ∈π−1(U) is a local chart onTM. It will be denoted by(π−1(U),φ =(xi ,yi)). The set of these “induced” local charts determines a differ-entiable structure onTM such thatπ : TM → M is a differentiablemanifold of dimension 2n and (TM,π,M) is a differentiable vectorbundle.

3

4 1 The Geometry of tangent manifold

A change of coordinate onM, (U,ϕ = xi)→ (V,ψ = xi) given by

xi = xi(x j), with rank

(∂ xi

∂x j

)= n has the corresponding change of

coordinates onTM : (π−1(U), Φ = (xi ,yi))→ (π−1(V),Ψ = (xi , yi)),(U ∩V 6= φ), given by:

xi = xi(x j), rank

(∂ xi

∂x j

)= n,

yi =∂ xi

∂x j yj .

(1.1.1)

The Jacobian ofΨΦ−1 is det

(∂ xi

∂x j

)2

> 0. So the manifoldTM is

orientable.The tangent spaceTuTM at a pointu ∈ TM to TM is a 2n−di-

mensional vector space, having the natural basis

∂

∂xi ,∂

∂yi

at u. A

change of coordinates (1.1.1) onTM implies the change of naturalbasis, at pointu as follows:

∂∂xi =

∂ x j

∂xi

∂∂ x j +

∂ y j

∂xi

∂∂ y j ,

∂∂yi =

∂ x j

∂xi

∂∂ y j .

(1.1.2)

A vector Xu ∈ TuTM is given byX = Xi(u)∂

∂xi +Yi(u)∂

∂yi . Then a

vector fieldX on TM is sectionX : TM → TTM of the projectionπ∗ : TTM→ TM. This isπ∗(x,y,X,Y) = (x,y).

From the last formula (1.1.2) we can see that

(∂

∂yi

)at point

u∈ TM span ann−dimensional vector subspaceV(u) of TuTM. ThemappingV : u ∈ TM → V(u) ⊂ TuTM is an integrable distributioncalled the vertical distribution. ThenVTM=

⋃

u∈TM

V(u) is a subbundle

of the tangent bundle(TTM,π∗(u),TM) to TM. As π : TM → M is asubmersion it follows thatπ∗,u : TuTM → Tπ(u)M is an epimorphism

1.1 The manifold TM 5

of linear spaces. The kernel ofπ∗,u is exactly the vertical subspacesV(u).

We denote byX v(TM) the set of all vertical vector field onTM. Itis a real subalgebra of Lie algebra of vector fields onTM, X (TM).

Consider the cotangent spaceT∗u TM, u∈TM. It is the dual of space

TuTM and(dxi ,dyi)u is the natural cobasis and with respect to (1.1.1)we have

dxi =∂ xi

∂x j dxj ,dyi =∂ xi

∂x j dyj +∂ yi

∂x j dxj . (1.1.3)

The almost tangent structure of tangent bundle is defined as

J =∂

∂yi ⊗dxi (1.1.4)

By means of (1.1.2) and (1.1.3) we can prove thatJ is globallydefined onTM and that we have

J

(∂

∂xi

)=

∂∂yi , J

(∂

∂yi

)= 0. (1.1.4’)

It follows that the following formulae hold:

J2 = JJ = 0,KerJ= ImJ=VTM.

The almost cotangent structureJ∗ is defined by

J∗ = dxi ⊗∂

∂yi .

Therefore, we obtain

J∗(dxi) = 0,J∗(dyi) = dxi .

The Liouville vector field onTM = TM\0 is defined by

C= yi ∂∂yi . (1.1.5)

It is globally defined onTM andC 6= 0.A smooth functionf : TM → R is calledr ∈ Z homogeneous with

respect to the variablesyi if, f (x,ay) = ar f (x,y), ∀a ∈ R+. The Eu-


ler theorem holds:A function f∈ F (TM) differentiable onTM is rhomogeneous with respect to yi if and only if

LC f = C f = yi ∂ f∂yi = r f , (1.1.6)

LC being the Lie derivation with respect toC.A vector fieldX ∈X (TM) is r−homogeneous with respect toyi if

LCX = (r −1)X, whereLCX = [C,X].Finally an 1-formω ∈X ∗(TM) is r−homogeneous ifLCω = rω .Evidently, the notion of homogeneity can be extended to a tensor

field T of type(r,s) on the manifoldTM.

1.2 Semisprays on the manifoldTM

The notion of semispray on the total spaceTM of the tangent bundleis strongly related with the second order differential equations on thebase manifoldM:

d2xi

dt2+2Gi

(x,

dxdt

)= 0. (1.2.1)

Writing the equation (1.2.1), onTM, in the equivalent form

dyi

dt2+2Gi(x,y) = 0,yi =

dxi

dt(1.2.2)

we remark that with respect to the changing of coordinates (1.1.1) onTM, the functionsGi(x,y) transform according to:

2Gi =∂ xi

∂x j 2G j −∂ yi

∂x j yj . (1.2.3)

But (1.2.2) are the integral curve of the vector field:

S= yi ∂∂xi −2Gi(x,y)

∂∂yi (1.2.4)

1.3 Nonlinear connections 7

By means of (1.2.3) one proves:S is a vector field globally definedon TM. It is called a semispray onTM andGi are the coefficients ofS.

S is homogeneous of degree 2 if and only if its coefficientsGi arehomogeneous functions of degree 2. IfS is 2-homogeneous then wesay thatS is a spray.

If the base manifoldM is paracompact, then onTM there existsemisprays.

1.3 Nonlinear connections

As we have seen in the first section of this chapter, the vertical dis-tributionV is a regular,n-dimensional, integrable distribution onTM.Then it is naturally to look for a complementary distribution ofVTM.It will be called a horizontal distribution. Such distribution is equiva-lent with a nonlinear connection.

Consider the tangent bundle(TM,π,M) of the base manifoldMand the tangent bundle(TTM,π∗,TM) of the manifoldTM. As weknow the kernel ofπ∗ is the vertical subbundle(VTM,πV ,TM). Itsfibres are the vertical spacesV(u), u∈ TM.

For a vector fieldX ∈X (TM), in local natural basis, we can write:

X = Xi(x,y)∂

∂xi +Yi(x,y)∂

∂yi .

ShorterX = (xi ,yi ,Xi,Yi).The mappingπ∗ : TTM→ TM has the local formπ∗(x,y,X,Y) =

(x,X). The points of the submanifoldVTM are of the form(x,y,0,Y).Let us consider the pull-back bundle

π∗(TM) = TM×πTM = (u,v) ∈ TM×TM|π(u) = π(v).

The fibres ofπ∗(TM), i.e., π∗u(TM) are isomorphic toTπ(u)M.

Then, we can define the following morphismπ ! : TTM−→ π∗(TM)by π!(Xu) = (u,π∗,u(Xu)). Therefore we have

kerπ! = kerπ∗ =VTM.

We can prove, without difficulties that the following sequence of vec-tor bundles overTM is exact:

0−→VTMi

−→TTMπ!−→π∗(TM)−→ 0 (1.3.1)


Thus, we can give

Definition 1.3.1.A nonlinear connection on the tangent manifoldTMis a left splitting of the exact sequence (1.3.1).

Consequently, a nonlinear connection onTM is a vector bundlemorphismC : TTM→VTM, with the property thatC i = 1VTM.

The kernel of the morphismC is a vector subbundle of the tangentbundle(TTM,π∗,TM) denoted by(NTM,πN,TM) and called thehorizontalsubbundle. Its fibresN(u) determine a regularn-dimensionaldistributionN : u∈ TM → N(u) ⊂ TuTM, complementary to the ver-tical distributionV : u∈ TM →V(u)⊂ TuTM i.e.

TuTM = N(u)⊕V(u),∀u∈ TM. (1.3.2)

Therefore, a nonlinear connection onTM induces the following Whit-ney sum:

TTM= NTM⊕VTM. (1.3.2’)

The reciprocal of this property is true, too.An adapted local basis to the direct decomposition (1.3.2) is of the

form

(δ

δxi ,∂

∂y j

)

u, where

δδxi =

∂∂xi −N j

i(x,y)∂

∂y j (1.3.3)

andδ

δxi

∣∣∣∣u, (i = 1, ...,n) are vector fields that belong toN(u).

They aren−linear independent vector fields and are independent

from the vector fields

(∂

∂yi

)

u(i = 1, ...,n) which belong toV(u).

The functionsN ji (x,y) are called the coefficients of the nonlinear

connection, denoted in the following byN.Remarking thatπ∗,u : TuTM → Tπ(u)M is an epimorphism the re-

striction ofπ∗,u to N(u) is an isomorphism fromN(u) soTπ(u)M. Sowe can take the inverse maplh,u the horizontal lift determined by thenonlinear connectionN.

Consequently, the vector fieldsδ

δxi

∣∣∣∣u

can be given in the form

(δ

δxi

)

u= lh,u

(∂

∂xi

)

π(u).


With respect to a change of local coordinates on the base manifold M

we have∂

∂xi =∂ x j

∂xi

θ∂ x j .

Consequently

(δ

δxi

)

uare changed, with respect to (1.1.1), in the

formδ

δxi =∂ x j

∂xi

δδ x j . (1.3.4)

It follows, from (1.3.3), that the coefficientsNij(x,y) of the nonli-

near connectionN, with respect to a change of local coordinates onthe manifoldTM, (1.1.1) are transformed by the rule:

∂ x j

∂xkNki = N j

k∂ xk

∂xi +∂ y j

∂xi . (1.3.5)

The reciprocal property is true, too.We can prove without difficulties that there exists a nonlinear con-

nection onTM if M is a paracompact manifold. The validity of thissentence is assured by the following theorem:

Theorem 1.3.1.If S is a semispray with the coefficients Gi(x,y) thenthe system of functions

Nij(x,y) =

∂Gi

∂y j (1.3.6)

is the system of coefficients of a nonlinear connection N.

Indeed, the formula (1.2.3) and∂

∂yi =∂ x j

∂xi

∂∂ y j give the rule of

transformation (1.3.5) forNij from (1.3.6).

The adapted dual basisdxi ,δyi of the basis

(δ

δxi ,∂

∂yi

)has the

1-formsδyi as follows:

δyi = dyi +Nijdxj . (1.3.7)

With respect to a change of coordinates, (1.1.1), we have

dxi =∂ xi

∂x j dxj , δ yi =∂ xi

∂x j δy j . (1.3.7’)


Now, we can consider the horizontal and vertical projectorsh andvwith respect to direct decomposition (1.3.2):

h=δ

δxi ⊗dxi , v=∂

∂yi ⊗δyi . (1.3.8)

Some remarkable geometric structures, as the almost product P andalmost complex structureF, are determined by the nonlinear connec-tion N:

P=δ

δxi ⊗dxi −∂

∂yi ⊗δyi = h−v. (1.3.9)

F=δ

δxi ⊗δyi −∂

∂yi ⊗dxi . (1.3.9’)

It is not difficult to see thatP andF are globally defined onTM and

PP= Id, FF=−Id. (1.3.10)

With respect to (1.3.2) a vector fieldX ∈ X (TM) can be uniquelywritten in the form

X = hX+vX = XH +XV (1.3.11)

with XH = hX andXV = vX.An 1-form ω ∈ X ∗(TM) has the similar form

ω = hω +vω ,

wherehω(X) = ω(XH), vω(X) = ω(XV).A d−tensor fieldT on TM of type (r,s) is called a distinguished

tensor field (shortly ad−tensor) if

T(ω1, ...,ωr ,X1, ...,Xs) = T(ε1ω1, ...,εrω r ,ε1X1, ...,εsXs)

whereε1, ...,εr , ... areh or v.ThereforehX = XH , vX = XV , hω = ωH , vω = ωV ared−vectors

or d−covectors. In the adapted basis

(δ

δxi ,∂

∂yi

)we have

XH = Xi(x,y)δ

δxi , XV = Xi ∂∂yi

and


ωH = ω j(x,y)dxj , ωV = ω jδy j .

A change of local coordinates onTM : (x,y) → (x, y) leads to thechange of coordinates ofXH ,XV,ωH ,ωV by the classical rules oftransformation:

Xi =∂ xi

∂x j Xj , ω j =

∂ xi

∂x j ω i etc.

So, ad−tensorT of type(r,s) can be written as

T = T i1...irj1... js

(x,y)δ

δxi1⊗ ...⊗

∂∂yir

⊗dxj1 ⊗ ...⊗δy js. (1.3.12)

A change of coordinates (1.1.1) implies the classical rule:

T i1...irj1... js

(x, y) =∂ xi1

∂xh1...

∂ xir

∂xhr

∂xk1

∂ x j1...

∂xks

∂ y js= Th1...h2

k1...ks. (1.3.12’)

Next, we shall study the integrability of the nonlinear connectionNand of the structuresP andF.

Since

(δ

δxi

)i = 1, ...,n is an adapted basis toN, according to the

Frobenius theorem it follows thatN is integrable if and only if the Lie

brackets

[δ

δxi ,δ

δx j

], i, j = 1, ...,n are vector fields in the distribution

N.But we have [

δδxi ,

δδx j

]= Rh

i j∂

∂yh (1.3.13)

where

Rhi j =

δNhi

δx j −δNh

j

δxi . (1.3.13’)

It is not difficult to prove thatRhi j are the components of ad−tensor

field of type (1,2). It is called thecurvaturetensor of the nonlinearconnectionN.

We deduce:

Theorem 1.3.2.The nonlinear connection N is integrable if and onlyif its curvature tensor Rhi j vanishes.

The weak torsionthi j of N is defined by


thi j =

∂Nhi

∂yi −∂Nh

j

∂yi . (1.3.14)

It is a d-tensor field of type(1,2), too. We say thatN is a symmetricnonlinear connection if its weak torsionth

i j vanishes.Now is not difficult to prove:

Theorem 1.3.3.1 The almost product structureP is integrable if andonly if the nonlinear connection N is integrable;

2 The almost complex structureF is integrable if and only if the non-linear connection N is symmetric and integrable.

The proof is simple using the Nijenhuis tensorsNP andNF. ForNPwe have the expression

NP(X,Y)=P2[X,Y]+[PX,PY]−P[PX,Y]−P[X,PY],∀X,Y∈ χ(TM).

Also we can see that each structureP orF characterizes the nonlin-ear connectionN.

Autoparallel curves of a nonlinear connectioncan be obtainedconsidering the horizontal curves as follows.

A curvec : t ∈ I ⊂ R→ (xi(t),yi(t)) ∈ TM has the tangent vector ˙cgiven by

c= cH + cV =dxi

dtδ

δxi +δyi

dt∂

∂yi (1.3.15)

whereδyi

dt=

dyi

dt+Ni

j(x,y)dxj

dt. (1.3.15’)

The curvec is ahorizontal curveif cV = 0 orδyi

dt= 0.

Evidently, if the functionsxi(t), t ∈ I are given andyi(t) are the so-lutions of this system of differential equations, then we have an hori-zontal curvexi = xi(t), yi = yi(t) in TM with respect toN.

In the caseyi =dxi

dt, the horizontal curves are called the autoparallel

curves of the nonlinear connectionN. They are characterized by thesystem of differential equations

dyi

dt+Ni

j(x,y)dxj

dt= 0, yi =

dxi

dt. (1.3.16)

1.4 N-linear connections 13

A theorem of existence and uniqueness can be easy formulatedforthe autoparallel curves of a nonlinear connectionN given by its coef-ficientsNi

j(x,y).

1.4 N-linear connections

An N-linear connection on the manifoldTM is a special linear con-nectionD on TM that preserves by parallelism the horizontal distri-butionN and the vertical distributionV. We study such linear connec-tions determining the curvature, torsion and structure equations.

Throughout this sectionN is an a priori given nonlinear connectionwith the coefficientsNi

j .

Definition 1.4.1.A linear connectionD on the manifoldTM is calledadistinguished connection(a d-connection for short) if it preserves byparallelism the horizontal distributionN.

Thus, we haveDh = 0. It follows that we have also:Dv = 0 andDP= 0.

If Y =YH +YV we get

DXY = (DXYH)H +(DXYV)V , ∀X,Y ∈ χ(TM).

We can easily prove:

Proposition 1.4.1.For a d-connection the following conditions areequivalent:

1o DJ = 0;2o DF= 0.

Definition 1.4.2.A d-connectionD is called anN-linear connectionifthe structureJ (orF) is absolute parallel with respect toD, i.e.DJ= 0.

In an adapted basis anN-linear connection has the form:

D δδxj

δδxi = Lh

i jδ

δxh ; D δδxj

∂∂yi = Lh

i j∂

∂yh ,

D ∂∂yj

δδxi =Ch

i jδ

δxh ; D ∂∂yj

∂∂yi =Ch

i j∂

∂yh .

(1.4.1)

The set of functionsDΓ = (Nij(x,y),L

hi j (x,y),C

hi j (x,y)) are called

the local coefficients of theN−linear connectionD. SinceN is fixed


we denote sometimes byDΓ (N)= (Lhi j (x,y),C

hi j (x,y)) the coefficients

of D.

For instance theBΓ (N) =

(∂Nh

i

∂y j ,0

)are the coefficient of a spe-

cial N−linear connection, derived only from the nonlinear connectionN. It is called theBerwald connectionof the nonlinear connectionN.

We can prove this showing that the system of functions

(∂Nh

i

∂y j

)has

the same rule of transformation, with respect to (1.1.1), asthe coef-ficientsLh

i j . Indeed, under a change of coordinates (1.1.1) onTM the

coefficients(Lhi j ,C

hi j ) are transformed by the rules:

Lhi j =

∂ xh

∂xsLspq

∂xp

∂ xi

∂xq

∂ x j −∂ 2xh

∂xp ∂xq

∂xp

∂ xi

∂xq

∂ x j ,

Chi j =

∂ xh

∂xsCspq

∂xp

∂ xi

∂xq

∂ x j .

(1.4.2)

So, the horizontal coefficientsLhi j of D have the same rule of trans-

formation of the local coefficients of a linear connection onthe basemanifold M. The vertical coefficientsCh

i j are the components of a(1,2)-type d-tensor field.

But, conversely, if the set of functions(Lijk(x,y),C

ijk(x,y)) are

given, having the property (1.4.2), then the equalities (1.4.1) deter-mine anN−linear connectionD onTM.

For anN−linear connectionD on TM we shall associate two op-erators ofh− andv− covariant derivation on the algebra ofd−tensorfields. For eachX ∈ χ(TM) we set:

DHXY = DXHY, DH

X f = XH f , ∀Y ∈ χ(TM),∀ f ∈ F (TM).(1.4.3)

If ω ∈ χ∗(TM), we obtain

(DHX ω)(Y) = XH(ω(Y))−ω(DH

XY). (1.4.3’)

So we may extend the action of the operatorDHX to any d-tensor field

by asking thatDHX preserves the type of d-tensor fields, isR-linear,

satisfies the Leibniz rule with respect to tensor product andcommuteswith the operation of contraction.DH

X will be called theh-covariantderivationoperator.

In a similar way, we set:


DVXY = DXVY, DV

X f = XV( f ), ∀Y ∈ χ(TM), ∀ f ∈ F (TM). (1.4.4)

andDV

Xω = XV(ω(Y))−ω(DVXY), ∀ω ∈ χ(TM).

Also, we extend the action of the operatorDVX to any d-tensor field

in a similar way as we did forDHX . DV

X is called thev-covariant ofderivation operator.

Consider now a d-tensorT given by (1.3.12). According to (1.4.1)its h−covariant derivation is given by

DHX T = XkT i1···ir

j1··· js|k

δδxi1

⊗·· ·⊗∂

∂yir⊗dxj1 ⊗·· ·⊗dxjs (1.4.5)

where

T i1···irj1··· js|k

=δT i1···ir

j1··· js

δxk +Li1pkT

pi2···irj1··· js

+ · · ·+LirpkT

i1···ir−1pj1··· js

−

−Lpj1kT

i1···irp j2··· js

−·· ·−Lpjsk

T i1···irj1··· js−1p.

(1.4.5’)

Thev-covariant derivativeDVXT is as follows:

DVXT = XkT i1···ir

j1··· js|k

δδxi1

⊗·· ·⊗∂

∂yir⊗dxj1 ⊗·· ·⊗dxjs, (1.4.6)

where

T i1···irj1··· js

|k =∂T i1···ir

j1··· js

∂yk +Ci1pkT

pi2···irj1··· js

+ · · ·+CirpkT

i1···ir−1pj1··· js

−

−Cpj1kT

i1···irp j2··· js

−·· ·−Cpjrk

T i1···irj1··· js−1p.

(1.4.6’)

For instance, ifg is a d-tensor of type (0,2) having the componentsgi j (x,y), we have

gi j |k =δgi j

δxk −Lpikgp j −Lp

jkgip,

gi j |k=∂gi j

∂yk −Cpikgp j −Cp

jkgip.

(1.4.7)


For the operators “|” and “|” we preserve the same denomination ofh− andv−covariant derivations.

The torsionT of anN-linear is given by:

T(X,Y) = DXY−DYX− [X,Y], ∀X,Y ∈ χ(TM). (1.4.8)

The horizontal parthT(X,Y) and the vertical onevT(X,Y), for X ∈XH,XV andY ∈ YH ,YV determine five d-tensor fields of torsionT:

hT(XH,YH) = DHXYH −DH

Y XH − [XH,YH ]H ,

vT(XH,YH) =−v[XH ,YH ]V ,

hT(XH,YV) =−DVYXH − [XH,YV ]V ,

vT(XH,YV) = DHXYV − [XH,YV ]V ,

vT(XV ,YV) = DVXYV −DV

YXV − [XV ,YV ]V .

(1.4.9)

With respect to the adapted basis, the components of torsionare asfollows:

hT

(δ

δxi ,δ

δx j

)= Tk

jiδ

δxk ;vT

(δ

δxi ,δ

δx j

)= Rk

ji∂

∂yk ;

hT

(∂

∂yi ,δ

δx j

)=Ck

jiδ

δxk ;

vT

(∂

∂yi ,δ

δx j

)= Pk

ji∂

∂yk ;

vT

(∂

∂yi ,∂

∂y j

)= Sk

ji∂

∂yk ,

(1.4.9’)

whereCijk are thev−coefficients ofD, Ri

jk is the curvature tensor ofthe nonlinear connectionN and

T ijk = Li

jk −Lik j,S

ijk =Ci

jk −Cik j,P

ijk =

∂Nij

∂yk −Lik j. (1.4.10)

TheN−linear connectionD is said to be symmetric ifT ijk = Si

jk =0.

Next, we study the curvature of anN-linear connectionD:


R(X,Y)Z=DXDYZ−DYDXZ−D[X,Y]Z, ∀X,Y,Z∈ χ(TM). (1.4.11)

As D preserves by parallelism the distributionN andV it follows thatthe operatorR(X,Y) = DXDY −DYDX −D[X,Y] carries the horizontalvector fieldsZH into horizontal vector fields and vertical vector fieldsinto verticals. Consequently we have the formula:

R(X,Y)Z = hR(X,Y)ZH +vR(X,Y)ZV , ∀X,Y,Z ∈ χ(TM).

SinceR(X,Y) =−R(Y,X), we obtain:

Theorem 1.4.1.The curvature R of a N-linear connection D on thetangent manifold TM is completely determined by the following sixd-tensor fields:

R(XH,YH)ZH = DHX DH

Y ZH −DHY DH

X ZH −D[XH ,YH ]ZH ,

R(XH,YH)ZV = DHX DH

Y ZV −DHY DH

X ZV −D[XH ,YH ]ZV ,

R(XV,YH)ZH = DVXDH

Y ZH −DHY DV

XZH −D[XV ,YH ]ZH ,

R(XV,YH)ZV = DVXDH

Y ZV −DHY DV

XZV −D[XV ,YH ]ZV ,

R(XV,YV)ZH = DVXDV

YZH −DVYDV

XZH −D[XV ,YV ]ZH ,

R(XV,YV)ZV = DVXDV

YZV −DVYDV

XZV −D[XV ,YV ]ZV .

(1.4.12)

As the tangent structure J is absolute parallel with respectto D wehave that

JR(X,Y)Z = R(X,Y)JZ.

ThenR(X,Y)Z has only three components

R(XH,YH)ZH ,R(XV,YH)ZH ,R(XV,YV)ZH .

In the adapted basis these are:


R

(δ

δxk ,δ

δx j

)δ

δxh = R ih k j

δδxi ;

R

(∂

∂yk ,δ

δx j

)δ

δxh = P ih k j

δδxi ;

R

(∂

∂yk ,∂

∂y j

)δ

δxh = S ih k j

δδxi .

(1.4.13)

The other three components are obtained by applying the operatorJ

to the previous ones. So, we haveR

(δ

δxh ,δ

δx j

)∂

∂yh = R ih jk

∂∂yh etc.

Therefore, the curvature of anN−linear connectionDΓ =(Nij ,L

ijk,C

ijk)

has only three local componentsR ih jk, P i

h jk andS ih jk. They are given

by

R ih jk =

δLih j

δxk −δLi

hk

δx j +Lsh jL

isk−Ls

hkLis j+Ci

hsRsjk;

P ih jk =

∂Lih j

∂yk −Cihk| j +Ci

hsPsjk;

S ih jk =

∂Cih j

∂yk −∂Ci

hk

∂y j +Csh jC

isk−Cs

hkCis j.

(1.4.14)

If Xi(x,y) are the components of ad-vector field onTM then from(1.4.12) we may derive the Ricci identities forXi with respect to anN-linear connectionD. They are:

Xi| j |k−Xi

|k| j = XsR is jk−Xi

|sTsjk −Xi|sRs

jk,

Xi| j |k−Xi|k| j = XsP i

s jk−Xi|sC

sjk −Xi |sPs

jk,

Xi| j |k−Xi |k| j = XsS is jk−Xi|sSs

jk.

(1.4.15)

We deduce some fundamental identities for theN−linear connec-tion D, applying the Ricci identities to the Liouville vector field

C= yi ∂∂yi . Considering the d-tensors

Dij = yi

| j , dij = yi | j (1.4.16)

1.5 Parallelism. Structure equations 19

calledh− andv−deflectiontensors ofD we obtain:

Theorem 1.4.2.For any N-linear connection D the following identi-ties hold:

Dik|h−Di

h|k = ysR is kh−Di

sTskh−di

sRskh,

Dik|h−di

h|k = ysP is kh−Di

sCskh−di

sPskh,

dik|h−di

h|k = ysS is kh−di

sSskh.

(1.4.17)

Others fundamental identities are the Bianchi identities,which areobtained by writing in the adapted basis the following Bianchi identi-ties:

Σ [DXT(Y,Z)−R(X,Y)Z+T(T(X,Y),Z)] = 0,

Σ [(DXR)(U,Y,Z)+R(T(X,Y),Z)U ] = 0,(1.4.18)

whereΣ means cyclic summation overX,Y,Z.

1.5 Parallelism. Structure equations

Let DΓ (N) = (Lijk,C

ijk) be anN-linear connection and the adapted

basis

(δ

δxi ,∂

∂yi

), i = 1,n.

As we know, a curvec : t ∈ I → (xi(t),yi(t))∈ TM, has the tangent

vector c = cH + cV given by (3.15), i.e. ˙c =dxi

dtδ

δxi +δyi

dt∂

∂yi . The

curvec is horizontal ifδyi

dt= 0, and it is an autoparallel curve with

respect to the nonlinear connectionN ifδyi

dt= 0, yi =

dxi

dt.

We set

DXdt

= DcX, DX =DXdt

dt, ∀X ∈ χ(TM). (1.5.1)

HereDXdt

is the covariant differential along the curvec with respect to

the theN-linear connectionD.


SettingX = XH +XV , XH = Xi δδxi , XV = Xi ∂

∂yi we have

DXdt

=DXH

dt+

DXV

dt=

=

Xi

|kdxk

dt+Xi|k

δyk

dt

δ

δxi +

Xi

|kdxk

dt+ Xi |k

δyk

dt

.

(1.5.2)

Consider theconnection 1-formsof D:

ω ij = Li

jkdxk+Cijkδyk. (1.5.3)

ThenDXdt

takes the form

DXdt

=

dXi

dt+Xsω i

s

dt

δ

δxi +

dXi

dt+ Xsω i

s

dt

∂

∂yi . (1.5.4)

The vector fieldX on TM is said to be parallel along the curve

c, with respect to theN-linear connectionD(N) ifDXdt

= 0. Using

(1.5.2), the equationDXdt

= 0 is equivalent toDXH

dt= 0,

DXV

dt= 0.

According to (1.5.4) we obtain:

Proposition 1.5.1.The vector field X= Xi δδxi + Xi ∂

∂yi from χ(TM)

is parallel along the parametrized curve c in TM, with respect to theN-linear connection DΓ (N) = (Li

jk,Cijk) if and only if its coefficients

Xi(x(t),y(t)) and Xi(x(t),y(t)) are solutions of the linear system ofdifferential equations

dZi

dt+Zs(x(t),y(t))

ωis(x(t),y(t))

dt= 0.

A theorem of existence and uniqueness for the parallel vector fieldsalong a curvec on the manifoldTM can be formulated on the classicalway.

Thehorizontal geodesicof D are the horizontal curvesc : I → TM

with the propertyDcc= 0. TakingXi =dxi

dt, Xi =

δyi

dt= 0 we get:

1.5 Parallelism. Structure equations 21

Theorem 1.5.1.The horizontal geodesics of an N-linear connectionare characterized by the system of differential equations:

d2xi

dt2+Li

jk(x,y)dxj

dtdxk

dt= 0,

dyi

dt+Ni

j(x,y)dxj

dt= 0. (1.5.5)

Now we can consider a curvecVx0

on the fibreTx0M = π−1(x0). It isrepresented by

xi = xi0, yi = yi(t), t ∈ I ,

cvx0

is called averticalcurve ofTM at the pointx0 ∈ M.Thevertical geodesicof D are the vertical curvescv

x0with the prop-

erty Dcvx0

cvx0= 0.

Theorem 1.5.2.The vertical geodesics at the point x0 ∈ M, of the N-linear connection DΓ (N) = (Li

jk,Cijk) are characterized by the fol-

lowing system of differential equations

xi = xi0,

d2yi

dt2+Ci

jk(x0,y)dyj

dtdyk

dt= 0. (1.5.6)

Obviously, the local existence and uniqueness of horizontal or ver-tical geodesics are assured if initial conditions are given.

Now, we determine the structure equations of anN-linear connec-tion D, considering the connection 1-formsω i

j , (1.5.3).First of all, we have:

Lemma 1.5.1.The exterior differential of1-formsδyi = dyi +Nijdxj

are given by:

d(δyi) =12

Rijsdxs∧dxj +Bi

jsδys∧dxj (1.5.7)

where

Bijk =

∂Nij

∂yk . (1.5.7’)

Remark 1.5.1. Bi jk are the coefficients of the Berwald connection.

Lemma 1.5.2.With respect to a change of local coordinate on themanifold TM, the following2-forms

d(dxi)−dxs∧ω is;d(δyi)−δys∧ω i

s


transform as the components of a d-vector field.The2-forms

dω ij −ωs

j ∧ω is

transform as the components of a d-tensor field of type(1,1).

Theorem 1.5.3.The structure equations of an N-linear connectionDΓ (N) = (Li

jk,Cijk) on the manifold TM are given by

d(dxi)−dxs∧ω is=−

(0)Ω i

d(δyi)−δys∧ω is=−

(1)Ω i

dω ij −ωs

j ∧ω is =−Ω i

j

(1.5.8)

where(0)Ω i and

(1)Ω i are the2-forms of torsion:

(0)Ω i =

12

T ijkdxj ∧dxk+Ci

jkdxj ∧δyk

(1)Ω i =

12

Rijkdxj ∧dxk+Pi

jkdxj ∧δyk+12

Sijkδy j ∧δyk

(1.5.9)

and the2-forms of curvatureΩ ij are given by

Ω ij =

12

R ij khdxk∧dxh+P i

j khdxk∧δyh+12

S ij khδy j ∧δyh. (1.5.10)

Proof. By means of Lemma (1.5.2), the general structure equations ofa linear connection onTM are particularized for anN-linear connec-tion D in the form (1.5.8). Using the connection 1-formsω i

j (1.5.3)

and the formula (1.5.7), we can calculate the forms(0)Ω i ,

(1)Ω i andΩ i

j .Then it is very easy to determine the structure equations (1.5.9).

Remark 1.5.2.The Bianchi identities of anN-linear connectionD canbe obtained from (1.5.8) by calculating the exterior differential of(1.5.8), modulo the same system (1.5.8) and using the exterior dif-

ferential of(0)Ω i

(1)Ω i andΩ i

j .

Chapter 2Lagrange spaces

The notion of Lagrange spaces was introduced and studied by thepresent author. The term “Lagrange geometry” is due to J. Kern, [121].We study the geometry of Lagrange spaces as a subgeometry of thegeometry of tangent bundle(TM,π,M) of a manifoldM, using theprinciples of Analytical Mechanics given by variational problem onthe integral of action of a regular Lagrangian, the law of conservation,Nother theorem etc. Remarking that the Euler - Lagrange equationsdetermine a canonical semispraySon the manifoldTM we study thegeometry of a Lagrange space using this canonical semi-spray Sandfollowing the methods given in the first chapter.

Beginning with the year 1987 there were published by author,aloneor in collaborations, some books on the Lagrange spaces and theHamilton spaces [174], and on the higher-order Lagrange andHamil-ton spaces [161], as well.

2.1 The notion of Lagrange space

First we shall define the notion of differentiable Lagrangian over thetangent manifoldTM andTM = TM\0, M being a realn−dimen-sional manifold.

Definition 2.1.1.A differentiable Lagrangian is a mappingL : (x,y)∈TM → L(x,y) ∈ IR, of classC∞ on TM and continuous on the nullsection 0 :M → TM of the projectionπ : TM → M.

The Hessian of a differentiable LagrangianL, with respect toyi , hasthe elements:

gi j =12

∂ 2L∂yi∂y j . (2.1.1)

23

24 2 Lagrange spaces

Evidently, the set of functionsgi j (x,y) are the components of ad-tensor field, symmetric and covariant of order 2.

Definition 2.1.2.A differentiable LagrangianL is calledregular if:

rank(gi j (x,y)) = n, onTM. (2.1.2)

Now we can give the definition of a Lagrange space:

Definition 2.1.3.A Lagrange space is a pairLn = (M,L(x,y)) formedby a smooth, realn-dimensional manifoldM and a regular LagrangianL(x,y) for which thed-tensorgi j has a constant signature over themanifoldTM.

For the Lagrange spaceLn = (M,L(x,y)) we say thatL(x,y) is thefundamental functionandgi j (x,y) is thefundamental(or metric) ten-sor.

Examples.

1Every Riemannian manifold(M,gi j (x)) determines a LagrangespaceLn = (M,L(x,y)), where

L(x,y) = gi j (x)yiy j . (2.1.3)

This example allows to say:If the manifold M is paracompact, then there exist LagrangiansL(x,y) such thatLn = (M,L(x,y)) is a Lagrange space.

2The following Lagrangian from electrodynamics

L(x,y) = mcγ i j (x)yiy j +

2em

Ai(x)yi +U (x), (2.1.4)

whereγ i j (x) is a pseudo-Riemannian metric,Ai(x) a covector field andU (x) a smooth function,m,c,e are physical constants, is a LagrangespaceLn. This is called the Lagrange space of electrodynamics.

We already have seen thatgi j (x,y) from (2.1.1) is a d-tensor field,i.e. with respect to (1.1.1), we have

gi j (x, y) =∂xh

∂ xi

∂xk

∂ x j ghk(x,y).

Now we can prove without difficulties:

Theorem 2.1.1.For a Lagrange space Ln the following propertieshold:

2.2 Variational problem. Euler-Lagrange equations 25

1The system of functions

pi =12

∂L∂yi

determines a d-covector field.2The functions

Ci jk =14

∂ 3L

∂yi∂y j∂yk =12

∂gi j

∂yk

are the components of a symmetric d−tensor field of type(0,3).3The 1-form

ω = pidxi =12

∂L∂yi dxi (2.1.5)

depend on the Lagrangian L only and are globally defined on themanifoldTM

4The2-formθ = dω = dpi ∧dxi (2.1.6)

is globally defined onTM and defines a symplectic structure onTM.

2.2 Variational problem. Euler-Lagrange equations

The variational problem can be formulated for differentiable La-grangians and can be solved in the case when the integral of actionis defined on the parametrized curves.

Let L : TM → R be a differentiable Lagrangian andc : t ∈ [0,1]→(xi(t))∈U ⊂ M be a smooth curve, with a fixed parametrization, hav-ing Imc⊂U , whereU is a domain of a local chart on the manifoldM.The curvec can be extended toπ−1(U)⊂ TM by

c : t ∈ [0,1]→ (xi(t),dxi

dt(t)) ∈ π−1(U).

So, Imc⊂ π−1(U).The integral of action of the LagrangianL on the curvec is given

by the functional:


I(c) =∫ 1

0L

(x,

dxdt

)dt. (2.2.1)

Consider the curves

cε : t ∈ [0,1]→ (xi(t)+ εV i(t)) ∈ M (2.2.2)

which have the same end pointsxi(0) andxi(1) as the curvec, V i(t) =V i(xi(t)) being a regular vector field on the curvec, with the propertyV i(0)=V i(1)= 0 andε is a real number, sufficiently small in absolutevalue, so that Imcε ⊂U .

The extension of a curvecε to TM is given by

cε : t ∈ [0,1] 7→

(xi(t)+ εV i(t),

dxi

dt+ ε

dVi

dt

)∈ π−1(U).

The integral of action of the LagrangianL on the curvecε is

I(cε) =

∫ 1

0L

(x+ εV,

dxdt

+ εdVdt

)dt. (2.2.1’)

A necessary condition forI(c) to be an extremal value ofI(cε) is

dI(cε)

dε|ε=0 = 0. (2.2.3)

Under our condition of differentiability, the operatorddε

is permut-

ing with the operator of integration.From (2.2.1’) we obtain

dI(cε)

dε=∫ 1

0

ddε

L(x+ εV,dxdt

+ εdVdt

)dt. (2.2.4)

But we have

ddε

L

(x+ εV,

dxdt

+ εdVdt

)|ε=0 =

∂L∂xi V

i +∂L∂yi

dVi

dt=

=

∂L∂xi −

ddt

∂L∂yi

V i +

ddt

∂L∂yi V

i

, yi =

dxi

dt.

2.2 Variational problem. Euler-Lagrange equations 27

Substituting in (2.2.4) and taking into account the fact that V i(x(t)) isarbitrary, we obtain the following theorem.

Theorem 2.2.1.In order for the functional I(c) to be an extremalvalue of I(cε) it is necessary for the curve c(t) = (xi(t)) to satisfythe Euler-Lagrange equations:

Ei(L) :=∂L∂xi −

ddt

∂L∂yi = 0, yi =

dxi

dt. (2.2.5)

For the Euler-Lagrange operatorEi =∂

∂xi −ddt

∂∂yi we have:

Theorem 2.2.2.The following properties hold true:1 Ei(L) is a d-covector field.2 Ei(L+L′) = Ei(L)+Ei(L′).3 Ei(aL) = aEi(L),a∈ R.

4 Ei

(dFdt

)= 0, ∀F ∈ F (TM) with

∂F∂yi = 0.

The notion ofenergy of a Lagrangian Lcan be introduced as in theTheoretical Mechanics [17], [164], by

EL = yi ∂L∂yi −L. (2.2.6)

We obtain, without difficulties:

Theorem 2.2.3.For every smooth curve c on the base manifold M thefollowing formula holds:

dEL

dt=−

dxi

dtEi(L), yi =

dxi

dt. (2.2.7)

Consequently:

Theorem 2.2.4.For any differentiable Lagrangian L(x,y) the energyEL is conserved along every solution curve c of the Euler-Lagrangeequations

Ei(L) = 0,dxi

dt= yi .

A Noether theorem can be proved:


Theorem 2.2.5.For any infinitesimal symmetry on M×R of the La-grangian L(x,y) and for any smooth functionφ(x) the following func-tion:

F (L,φ) =V i ∂L∂yi − τEL−φ (x)

is conserved on every solution curve c of the Euler-Lagrangeequa-

tions Ei(L) = 0, yi =dxi

dt.

Remark. An infinitesimal symmetry onM×R is given byx′i = xi +εV i(x, t), t ′ = t + ετ(x, t).

2.3 Canonical semispray. Nonlinear connection

Now we can apply the previous theory in order to study the LagrangespaceLn = (M,L(x,y)). As we shall see thatLn determines a canon-ical semispraySandSgives a canonical nonlinear connection on themanifoldTM.

As we know, the fundamental tensorgi j of the spaceLn is nonde-generate, andEi(L) is ad-covector field, so the equationsgi j E j(L) = 0have a geometrical meaning.

Theorem 2.3.1.If Ln = (M,L) is a Lagrange space, then the systemof differential equations

gi j E j(L) = 0, yi =dxi

dt(2.3.1)

can be written in the form:

d2xi

dt2+2Gi

(x,

dxdt

)= 0,yi =

dxi

dt(2.3.1’)

where

2Gi(x,y) =12

gi j

∂ 2L

∂y j∂xk yk−∂L∂x j

. (2.3.2)

Proof. We have

Ei(L) =∂L∂xi −

∂ 2L

∂yi∂xk +2gi jdyj

dt

, yi =

dxi

dt.

2.3 Canonical semispray. Nonlinear connection 29

So, (2.3.1) implies (2.3.1’), (2.3.2).The previous theorem tells us that the Euler Lagrange equations for

a Lagrange space are given by a system ofn second order ordinarydifferential equations. According with theory from Section 1.2, Ch. 1,it follows that the equations (2.3.1) determine a semispraywith thecoefficientsGi(x,y):


∂∂yi . (2.3.3)

S is called the canonical semispray of the Lagrange spaceLn.By means of Theorem 1.3.1, it follows:

Theorem 2.3.2.Every Lagrange space Ln = (M,L) has a canonicalnonlinear connection N which depends only on the fundamental func-tion L. The local coefficients of N are given by

Nij =

∂Gi

∂y j =14

∂∂y j

gik(

∂ 2L∂yk∂xhyh−

∂L∂xk

). (2.3.4)

Evidently:

Proposition 2.3.1.The canonical nonlinear connection N is symmet-

ric, i.e. tijk =∂Ni

j

∂yk −∂Ni

k

∂y j = 0.

Proposition 2.3.2.The canonical nonlinear connection N is invariantwith respect to the Caratheodory transformation

L′(x,y) = L(x,y)+∂ϕ(x)

∂xi yi . (2.3.5)

Indeed, we have

Ei(L′) = Ei

(L(x,y)+

dϕdt

)= Ei(L).⊓⊔

So,Ei(L′(x,y))= 0 determines the same canonical semispray as theone determined byEi(L(x,y)) = 0. Thus, the Caratheodory transfor-mation (2.3.5) preserves the nonlinear connectionN.

Example.The Lagrange space of electrodynamics,Ln = (M,L(x,y)),whereL(x,y) is given by (2.1.4) withU(x) = 0 has the canonicalsemispray with the coefficients:


Gi(x,y) =12

γ ijk(x)y

jyk−gi j (x)Fjk(x)yk, (2.3.6)

whereγ ijk(x) are the Christoffel symbols of the metric tensorgi j (x) =

mcγ i j (x) of the spaceLn andFjk is the electromagnetic tensor

Fjk(x,y) =e

2m

(∂Ak

∂x j −∂A j

∂xk

). (2.3.7)

Therefore, the integral curves of the Euler-Lagrange equation aregiven by the solution curves of theLorentz equations:

d2xi

dt2+ γ i

jk(x)dxj

dtdxk

dt= gi j (x)Fjk(x)

dxk

dt. (2.3.8)

According to (2.3.4), the canonical nonlinear connection of the La-grange space of electrodynamicsLn has the local coefficients givenby

Nij(x,y) = γ i

jk(x)yk−gik(x)Fk j(x). (2.3.9)

It is remarkable that the coefficientsNij from (2.3.9) are linear with

respect toyi .

Proposition 2.3.3.The Berwald connection of the canonical nonlin-ear connection N has the coefficients BΓ (N) = (γ i

jk(x),0).

Proposition 2.3.4.The solution curves of the Euler-Lagrange equa-tions and the autoparallel curves of the canonical nonlinear connec-tion N are given by the Lorentz equations(2.3.8).

In the last part of this section, we underline the following theorem:

Theorem 2.3.3.The autoparallel curves of the canonical nonlinearconnection N are given by the following system:

d2xi

dt2+Ni

j

(x,

dxdt

)dxj

dt= 0,

where Nij are given by(2.3.4).

2.4 Hamilton-Jacobi equations 31

2.4 Hamilton-Jacobi equations

Consider a Lagrange spaceLn = (M,L(x,y)) andN(Nij) its canonical

nonlinear connection. The adapted basis

(δ

δxi ,∂

∂yi

)to the horizontal

distributionN and the vertical distributionV has the horizontal vectorfields:

δδxi =

∂∂xi −N j

i∂

∂y j . (2.4.1)

Its dual is(dxi ,δyi), with

δyi = dyi +Nijdxj . (2.4.2)

Theorem 1.1.1 give us the momenta

pi =12

∂L∂yi , (2.4.3)

the 1-formω = pidxi (2.4.4)

and the 2-formθ = dω = dpi ∧dxi . (2.4.5)

These geometrical object fields are globally defined onTM. θ is asymplectic structureon the manifoldTM.

Proposition 2.4.5.In the adapted basis the 2-formθ is given by

θ = gi j δyi ∧dxj . (2.4.6)

Indeed,θ = dpi ∧ dxi =12

(δ

δxs

∂L∂yi dxs+

∂∂ys

∂L∂yi δys

)∧ dxi =

14

(δ

δxs

∂L∂yi −

δδxi

∂L∂ys

)dxs∧dxi +gisδys∧dxi .

But is not difficult to see that the coefficient ofdxs∧dxi vanishes.The triple(TM,θ ,L) is called a Lagrangian system.The energyEL of the spaceLn is given by (2.2.6). DenotingH =

12

EL, L =12

L, then (2.2.6) can be written as:

H = piyi −L (x,y). (2.4.7)


But, along the integral curve of the Euler-Lagrange equations (2.2.5)we have

∂H

∂xi =−∂L

∂xi =−dpi

dt.

And from (2.4.7), we get

∂H

∂ pi= yi =

dxi

dt.

So, we obtain:

Theorem 2.4.1.Along to integral curves of the Euler-Lagrange equa-tions the Hamilton-Jacobi equations:

dxi

dt=

∂H

∂ pi;

dpi

dt=−

∂H

∂xi , (2.4.8)

whereH is given by(2.4.7) and pi =12

∂L∂yi , are satisfied.

These equations are important in applications.

Example. For the Lagrange space of Electrodynamics with the fun-damental functionL(x,y) from (2.1.4) andU(x) = 0 we obtain

H =1

2mcγ i j (x)pi p j −

emc2Ai(x)pi +

e2

2mc3Ai(x)Ai(x)

(Ai = γ i j A j).Then, the Hamilton - Jacobi equations can be written withoutdiffi-

culties.Now we remark thatθ being a symplectic structure onTM, exterior

differentialdθ vanishes. But in adapted basis

dθ = dgi j ∧δyi ∧dxj +gi j dδyi ∧dxj = 0

reduces to:

12

(δgi j

δxk −δgik

δx j

)δyi∧dxj ∧dxk+

12

(∂gi j

∂xk −∂gk j

∂yi

)δyk∧δ yi∧dxj+

+gi j

(12

Rirsdxs∧dxr +Bi

rsδys∧dxr)∧dxj = 0.

2.5 MetricalN-linear connections 33

We obtain

Theorem 2.4.2.For any Lagrange space Ln the following identitieshold

gi j‖k−gik‖ j = 0, gi j‖k−gik‖ j = 0. (2.4.9)

Indeed, taking into account theh− andv−covariant derivations of

the metricgi j with respect to Berwald connectionBΓ (N)=

(∂Ni

j

∂yk ,0

),

i.e.

gi j‖k =δgi j

δxk −Brikgk j −Br

jkgir

andgi j‖k =∂gi j

∂yk , according with the properties∂gi j

∂yk = 2Ci jk , Bijk =

Bik j, we obtain (2.4.9).

2.5 Metrical N-linear connections

Let N(Nij) be the canonical nonlinear connection of the Lagrange

spaceLn = (M,L) andD anN−linear connection with the coefficientsDΓ (N) = (Li

jk,Cijk). Then, theh− andv− covariant derivations of the

fundamental tensorgi j , gi j |k andgi j |k are given by (1.4.7).Applying the theory ofN−linear connection from Chapter 1, one

proves without difficulties, the following theorem:

Theorem 2.5.1.1On the manifoldTM there exist only one N−linearconnection D which verifies the following axioms:

A1 N is canonical nonlinear connection of the space Ln.A2 gi j |k = 0 (D is h-metrical);A3 gi j |k = 0 (D is v-metrical);A4 T i

jk = 0 (D is h-torsion free);

A5 Sijk = 0 (D is v-torsion free).

2 The coefficients DΓ (N) = (Lijk,C

ijk) of D are expressed by the

following generalized Christoffel symbols:


Lijk =

12

gir

(δgrk

δx j +δgr j

δxk −δg jk

δxr

)

Cijk =

12

gir

(∂grk

∂y j +∂gr j

∂yk −∂g jk

∂yr

) (2.5.1)

3This connection depends only on the fundamental function L(x,y)of the Lagrange space Ln.

TheN-linear connectionD given by the previous theorem is calledthecanonical metric connectionand denoted byCΓ (N) = (Li

jk,Cijk).

By means of§1.5, Ch. 1, the connection 1-formsω ij of theCΓ (N)

areω i

j = Lijkdxk+Ci

jkδyk, (2.5.2)

Theorem 2.5.2.The canonical metrical connection CΓ (N) satisfiesthe following structure equations:

d(dxi)−dxk∧ω ik =−

(0)Ω i ,

d(δyi)−δyk∧ω ik =−

(1)Ω i ,

dω ij −ωk

j ∧ω ik =−Ω i

j

(2.5.3)

where 2-forms of torsion(0)Ω i and

(1)Ω i are as follows

(0)Ω i =Ci

jkdxj ∧δyk,

(1)Ω i =

12

Rijkdxj ∧dxk+Pi

jkdxj ∧δyk

(2.5.4)

and the 2-forms of curvatureΩ ij are

Ω ij =

12

R ij khdxk∧dxh+P i

j khdxk∧δyh+12

S ij khδyk∧δyh. (2.5.5)

Thed-tensors of torsionR ijk, P i

jk are given by (1.3.13’) and (1.4.10),

and the d-tensors of curvatureR ij kh, P i

j kh, S ij kh have the expressions

(1.4.14).

2.5 MetricalN-linear connections 35

Starting from the canonical metrical connectionCΓ (N)= (Lijk,C

ijk)

we can derive otherN-linear connections depend only on the spaceLn:

Berwald connectionBΓ (N) =

(∂Ni

j

∂yk ,0

); Chern-Rund connection

RΓ (N)= (Lijk,0) and Hashiguchi connectionHΓ (N)=

(∂Ni

j

∂yk ,Cijk

).

For special transformations of these connections, the following com-mutative diagram holds:

RΓ (N)ր ց

CΓ (N) −→ BΓ (N)ց ր

HΓ (N)

Some properties of the canonical metrical connectionCΓ (N) aregiven by:

Proposition 2.5.6.We have:

1∑(i jk)Ri jk = 0, (Ri jk = gihRhjk).

2Pi jk = gihPhjk is totally symmetric.

3The covariant curvature d-tensors Ri jkh = g jr R ri kh, Pi jkh = g jr P r

i khand Si jkh = gir S r

i kh are skew-symmetric with respect to the first twoindices.

4Si jkh =CiksCsjh−CihsCs

jk.5Cikh = gisCs

jh.

These properties can be proved using the propertydθ = 0, withθ = gi j δyi ∧dxj , the Ricci identities applied to the fundamental tensorgi j and the equationsgi j |k = 0, gi j |k = 0.

By the same method we can study the metrical connections withapriori givenh− andv− torsions.

Theorem 2.5.3.1There exists only one N-linear connectionDΓ (N)=(Li

jk,Cijk) which satisfies the following axioms:

A′1 N is canonical nonlinear connection of the space Ln,

A′2 gi j |k = 0 (D is h-metrical),

A′3 gi j |k = 0 (D is v-metrical),

A′4 The h-tensor of torsionT i

jk is a priori given.

A′5 The v-tensor of torsionSi

jk is a priori given.


2 The coefficients(Lijk,C

ijk) of D are given by

Lijk = Li

jk +12

gih(g jr Trkh+gkrTr

jh−ghrTrk j),

Cijk =Ci

jk +12

gih(g jr Srkh+gkrSr

jh−ghrSrk j)

(2.5.6)

where(Lijk,C

ijk) are the coefficients of the canonical metrical connec-

tion.

From now onT ijk, S

ijk will be denoted byT i

jk,Sijk and theN−linear

connection given by the previous theorem will be calledmetricalN−connectionof the Lagrange spaceLn.

Some particular cases can be studied using the expressions of thecoefficientsLi

jk andCijk. For instance the semi-symmetric case will be

obtained takingT ijk = δ i

jσk−δ ikσ j , Si

jk = δ ijτk−δ i

kτ j .

Proposition 2.5.7.The Ricci identities of the metrical N-linear con-nection DΓ (N) are given by:

Xi| j |k−Xi

|k| j = XrR ir jk −Xi

|rTr

jk −Xi |rRrjk,

Xi| j |k−Xi|k| j = XrP i

r jk −Xi|rC

rjk −Xi |rPr

jk,

Xi| j |k−Xi|k| j = XrS ir jk −Xi|rSr

jk.

(2.5.7)

Of course these identities can be extended to ad-tensor field of type(r,s).

DenotingDi

j = yi| j , di

j = yi | j . (2.5.8)

we have theh− andv− deflection tensors. They have the known ex-pressions:

Dij = ysLi

s j−Nij ; di

j = δ ij +ysCi

s j. (2.5.7’)

According to Ricci identities (2.5.7) we obtain:Theorem 2.5.4.For any metrical N-linear connection the followingidentities hold:

Dij |k−Di

k| j = ysR is jk−Di

sTsjk −di

sRsjk,

Dij |k−di

k| j = ysP is jk−Di

sCsjk −di

sPsjk,

dij |k−di

k| j = ysS is jk−di

sSsjk.

(2.5.9)

2.6 The electromagnetic and gravitational fields 37

We will apply this theory in a next section taking into account thecanonical metrical connectionCΓ (N) and takingT i

jk = 0, Sijk = 0.

Of course the theory of parallelism of vector fields and theh−geo-desics orv−geodesics for the metrical connectionN−linear connec-tions can be obtained as a consequence of the corresponding theoryfrom Ch. 1.

2.6 The electromagnetic and gravitational fields

Let us consider a Lagrange spacesLn = (M,L) endowed with thecanonical nonlinear connectionN and with the canonical metricalN−connectionCΓ (N) = (Li

jk,Cijk).

The covariant deflection tensorsD ji and d ji are given byDi j =gisDs

j , di j = gisdsj . We have:

Di j |k = gisDs

j |k, di j |k = gisds

j |k

etc. So, we have

Proposition 2.6.8.The covariant deflection tensors Di j and di j of thecanonical metrical N-connection CΓ (N) satisfy the identities:

Di j |k−Dik| j = ysRsi jk−disRsjk,

Di j |k−dik| j = ysPsi jk−DisCsjk −disPs

jk,

di j |k−dik| j = ysSsi jk.

(2.6.1)

The Lagrangian theory of electrodynamics lead us to introduce[182], [184], [183], [175]:

Definition 2.6.1.The d-tensor fields:

Fi j =12(Di j −D ji), fi j =

12(di j −d ji ) (2.6.2)

are theh- andv-electromagnetic tensorof the Lagrange spaceLn =(M,L).

The Bianchi identities forCΓ (N) and the identities (2.6.1) lead tothe following important result:

Theorem 2.6.1.The following generalized Maxwell equations hold:


Fi j |k+Fjk|i +Fki| j =− ∑(i jk)

CiosRs

jk,

Fi j |k+Fjk|i +Fki| j = 0,

(2.6.3)

where Cios=Ci jsy j , and ∑(i jk)

means cyclic sum.

Corollary 2.6.1. If the canonical nonlinear connection N of the spaceLn is integrable then the equations (2.6.3) reduce to:

∑(i jk)

Fi j |k = 0, ∑(i jk)

Fi j |k = 0. (2.6.3’)

If we putF i j = gisg jr Fsr (2.6.4)

andhJi = F i j

| j , vJi = F i j | j , (2.6.5)

then one can prove:

Theorem 2.6.2.The following laws of conservation hold:

hJi|i =

12F i j (Ri j −Rji )+F i j |rRr

i j,

vJi |i = 0,

(2.6.6)

where Ri j is the Ricci tensor Rhi jh .

Remark 2.6.1.In the Lagrange space of electrodynamics the tensorFjkis given by (2.3.6). The previous theory one reduces to the classicaltheory. NamelyFi j (x) satisfy the Maxwell equations∑

(i jk)

Fi j |k = 0 and

Fi j |k = 0, h ji|i = 0, v ji = 0.

Now, considering the lift toTM of the fundamental tensorgi j (x,y)of the spaceLn, given by

G= gi j dxi ⊗dxj +gi j δyi ⊗δy j

we can obtain the Einstein equations of the canonical metricconnec-tionCΓ (N). The curvature Ricci and scalar curvatures:

2.6 The electromagnetic and gravitational fields 39

Ri j = R hi jh ,Si j = Sh

i jh,′Pi j = P h

i jh,′′P h

i h j

R= gi j Ri j ,S= gi j Si j .(2.6.7)

Let us denote byHT i j ,

VT i j ,

1T i j and

2T i j the components in adapted

basis

(δ

δxi ,∂

∂yi

)of the energy momentum tensor on the manifold

TM.Thus we obtain:

Theorem 2.6.3.1The Einstein equations of the Lagrange space Ln=(M,L(x,y)) with respect to the canonical metrical connectionCΓ (N) = (Li

jk,Cijk) are as follows:

Ri j −12

Rgi j = κHT i j ,

′Pi j = κ1T i j

Si j −12

Sgi j = κVT(i)( j),

′′Pi j = κ2T i

j ,

(2.6.8)

whereκ is a real constant.

2The energy momentum tensorsHT i j and

VT i j satisfy the following

laws of conservation

κHT i j=−

12(Pih

jsRshi +2Rs

i j Pis),κ

VT

i

j |i = 0. (2.6.9)

The physical background of the previous theory is discussedbySatoshy Ikeda in the last chapter of the book [116].

The previous theory is very simple in the particular LagrangespacesLn havingP h

i jk = 0.We have:

Corollary 2.6.2. 1 If the canonical metrical connection CΓ (N) hasthe property Pi

j kh = 0, then the Einstein equations are

Ri j −12

Rgi j = κHT i j ,Si j −

12

Sgi j = κVT(i)( j) (2.6.10)

2The following laws of conservation hold:


HT

i

j |i= 0,VT

i

j |i = 0.

Remark 2.6.2.The Lagrange space of Electrodynamics,Ln, hasCΓ (N) = (γ i

jk(x),0), P ij kh = 0, S i

j kh = 0. The Einstein equations(2.6.10) reduce to the classical Einstein equations of the spaceLn.

2.7 The almost Kahlerian model of a Lagrange spaceLn

A Lagrange spaceLn = (M,L) can be thought as an almost Kahlerspace on the manifoldTM = TM\0, called the geometrical modelof the spaceLn.

As we know from section 3, Ch. 1 the canonical nonlinear connec-tion N determines an almost complex structureF(TM), expressed in(1.3.9’). This is

F=δ

δxi ⊗δyi −∂

∂yi ⊗dxi . (2.7.1)

F is integrable if and only ifRijk = 0.

F is globally defined onTM and it can be considered as aF (TM)−

linear mapping fromχ(TM) to χ(TM):

F

(δ

δxi

)=−

∂∂yi , F

(∂

∂yi

)=

δδxi ,(i = 1, ...,n). (2.7.1’)

The lift of the fundamental tensorgi j of the spaceLn with respectto N is defined by

G= gi j dxi ⊗dxj +gi j δyi ⊗δy j . (2.7.2)

EvidentlyG is a (pseudo-)Riemannian metric on the manifoldTM.The following result can be proved without difficulties:

Theorem 2.7.1.1The pair(G,F) is an almost Hermitian structureon TM, determined only by the fundamental function L(x,y) of Ln.

2The almost symplectic structure associated to the structure (G,F)is given by

θ = gi j δyi ∧dxj . (2.7.3)

2.7 The almost Kahlerian model of a Lagrange spaceLn 41

3The space(TM,G,F) is almost Kahlerian.

Indeed:1N,G,F are determined only byL(x,y).

We haveG(FX,FY) =G(X,Y), ∀X,Y ∈ χ(TM).2 In the adapted basisθ(X,Y) =G(FX,Y) is (2.7.3).3Taking into account (2.1.1), it follows thatθ is a symplectic struc-

ture (i.e.dθ = 0).

The spaceH2n = (TM,G,F) is calledthe almost Kahlerian modelof the Lagrange spaceLn. It has a remarkable property:

Theorem 2.7.2.The canonical metrical connection D with coeffi-cients CΓ (N) = (Li

jk,Cijk) of the Lagrange space Ln is an almost

Kahlerian connection, i.e.

DG= 0, DF= 0. (2.7.4)

Indeed, (2.7.1), (2.7.2), in the adapted basis imply (2.7.4).We can use this geometrical model to study the geometry of La-

grange spaceLn. For instance, the Einstein equations of the (pseudo)Riemannian space(TM,G) equipped with the metrical canonical con-nectionCΓ (N) are the Einstein equations of the Lagrange space stud-ied in the previous section of this chapter.

G.S. Asanov showed [27] that the metricG given by the lift (2.7.2)does not satisfies the principle of the Post-Newtonian calculus becausethe two terms ofG have not the same physical dimensions. This is thereason to introduce a new lift which can be used in a gauge theory ofphysical fields.

Let us consider the scalar field:

E = ||y||2 = gi j (x,y)yiy j . (2.7.5)

ε is called the absolute energy of the Lagrange spaceLn.We assume||y||2 > 0 and consider the following lift of the funda-

mental tensorgi j :

0G= gi j dxi ⊗dxj +

a2

||y||2gi j δyi ⊗δy j (2.7.6)

wherea> 0 is a constant.Let us consider also the tensor field onTM:

0F=−

||y||a

∂∂yi ⊗dxj +

a||y||

δδxi ⊗δyi , (2.7.7)


and 2-form0θ=

a||y||

θ , (2.7.8)

whereθ is from (2.7.3).We can prove:

Theorem 2.7.3.1The pair(0G,

0F) is an almost Hermitian structure

on the manifoldTM, depending only on the the fundamental func-tion L(x,y) of the space Ln.

2The almost symplectic structure0θ associated to the structure(

0G,

0F)

is given by(2.7.8).

30θ being conformal to symplectic structureθ , the pair(

0G,

0F) is con-

formal to the almost Kahlerian structure(G,F).

2.8 Generalized Lagrange spaces

A first natural generalization of the notion of Lagrange space is pro-vided by a notion which we call a generalized Lagrange space.Thisnotion was introduced by author in the paper [183].

Definition 2.8.1.A generalized Lagrange space is a pairGLn = (M,

gi j (x,y)), wheregi j (x,y) is a d-tensor field on the manifoldTM, oftype (0,2), symmetric, of rankn and having a constant signature onTM.

We continue to callgi j (x,y) thefundamentaltensor onGLn.

One easily sees that any Lagrange spaceLn = (M,L(x,y)) is a ge-neralized Lagrange space with the fundamental tensor

gi j (x,y) =12

∂ 2L(x,y)∂yi∂y j . (2.8.1)

But not any spaceGLn is a Lagrange spaceLn.Indeed, ifgi j (x,y) is given, it may happen that the system of partial

differential equations (2.8.1) does not admits solutions in L(x,y).

Proposition 2.8.9.1A necessary condition in order that the system

(2.8.1) admit a solution L(x,y) is that the d-tensor field∂gi j

∂yk =

2Ci jk be completely symmetric.

2.8 Generalized Lagrange spaces 43

2 If the condition1 is verified and the functions gi j (x,y) are 0-homogeneous with respect to yi ; then the function

L(x,y) = gi j (x,y)yiy j +Ai(x)y

i +U(x) (2.8.2)

is a solution of the system of partial differential equations (2.8.1)for any arbitrary d-covector field Ai(x) and any arbitrary functionU(x), on the base manifold M.

The proof of previous statement is not complicated.In the case when the system (2.8.1) does not admit solutions in the

functionsL(x,y) we say that the generalized Lagrange spaceGL2 =(M,gi j (x,y)) is not reducible to a Lagrange space.

Remark 2.8.3.The Lagrange spacesLn with the fundamental function(2.8.2) give us an important class of Lagrange spaces which includesthe Lagrange space of electrodynamics.

Examples.

1The pairGLn = (M,gi j ) with the fundamental tensor field

gi j (x,y) = e2σ (x,y)γ i j (x) (2.8.3)

whereσ is a function on(TM) andγ i j (x) is a pseudo-Riemannianmetric on the manifoldM is a generalized Lagrange space if the

d-covector field∂σ∂yi no vanishes.

It is not reducible to a Lagrange space. R. Miron and R. Tavakol[183] proved thatGLn = (M,gi j (x,y)) defined by (2.8.3) satisfiesthe Ehlers - Pirani - Schild’s axioms of General Relativity.

2The pairGLn = (M,gi j (x,y)), with

gi j (x,y) = γ i j (x)+

(1−

1n2(x,y)

)yiy j , yi = γ i j (x)y

j (2.8.4)

whereγ i j (x) is a pseudo-Riemannian metric andn(x,y) > 1 is asmooth function (n is a refractive index) give us a generalizedLagrange spaceGLn which is not reducible to a Lagrange space.This metric has been called by R. G. Beil, [40] the Miron’s metricfrom Relativistic Optics.

The restriction of the fundamental tensorgi j (x,y) (2.8.4) to a sec-tion SV : xi = xi ,yi =V i(x), (V i being a vector field) of the projection


π : TM → M, is given bygi j (x,V(x)). It gives us the known Synge’smetric tensor of the Relativistic Optics [243].

For a generalized Lagrange spaceGLn = (M,gi j (x,y)) an importantproblem is to determine a nonlinear connection obtained from the fun-damental tensorgi j (x,y). In the particular cases, given by the previoustwo examples this is possible. But, generally no.

We point out a method of determining a nonlinear connectionN,strongly connected to the fundamental tensorgi j of the spaceGLn, ifsuch kind of nonlinear connection exists.

Consider theabsolute energyE (x,y) of spaceGLn:

ε(x,y) = gi j (x,y)yiy j (2.8.5)

ε(x,y) is a Lagrangian.The Euler - Lagrange equations ofε(x,y) are

∂ε∂xi −

ddt

∂ε∂yi = 0, yi =

dxi

dt. (2.8.6)

Of course, according the general theory, the energyEε of the La-

grangianE (x,y), is Eε = yi ∂ε∂yi −E and it is preserved along the inte-

gral curves of the differential equations (2.8.6).If E (x,y) is a regular Lagrangian - we say that the spaceGLn is

weakly regular - it follows that the Euler-Lagrange equations deter-mine a semispray with the coefficients

2Gi(x,y) =∨g

is(

∂ 2ε∂ys∂x j y

j −∂ε∂xs

),

(∨gi j=

12

∂ 2ε∂yi∂y j

). (2.8.7)

Consequently, the nonlinear connectionN with the coefficients

Nij =

∂Gi

∂y j is determined only by the fundamental tensorgi j (x,y) of

the spaceGLn.In the case when we can not derive a nonlinear connection fromthe

fundamental tensorgi j , we give a priori a nonlinear connectionN andstudy the geometry of pair(GLn,N) by the methods of the geometryof Lagrange spaceLn.

For instance, using the adapted basis

(δ

δxi ,∂

∂yi

)to the distribu-

tionsN andV, respectively and its dual(dxi ,δyi) we can liftgi j (x,y)

2.8 Generalized Lagrange spaces 45

to TM:

G(x,y) = gi j (x,y)dxi ⊗dxj +a

‖y‖2gi j (x,y)δyi ⊗δy j

and can consider the almost complex structure

F=−‖y‖a

∂∂yi ⊗dxi +

a‖y‖

δδxi ⊗δyi

with ε(x,y)> 0 and‖y‖= ε1/2(x,y).The space(TM,F) is an almost Hermitian space geometrical asso-

ciated to the pair(GLn,N).J. Silagy, [241], make an exhaustive and interesting study of this

difficult problem.

Chapter 3Finsler Spaces

An important class of Lagrange Spaces is provided by the so-calledFinsler spaces.

The notion of Finsler space was introduced by Paul Finsler in1918and was developed by remarkable mathematicians as L. Berwald [42],E. Cartan [56], H. Buseman [52], H. Rund [218], S.S. Chern [60], M.Matsumoto [144], and many others.

This notion is a generalization of Riemann space, which gives animportant geometrical framework in Physics, especially inthe geo-metrical theory of physical fields, [18], [116], [129], [246], [252].

In the last 40 years, some remarkable books on Finsler geometryand its applications were published by H. Rund, M. Matsumoto, R.Miron and M. Anastasiei, A. Bejancu, Abate-Patrizio, D. Bao, S.S.Chern and Z. Shen, P. Antonelli, R. Ingarden and M. Matsumoto,R. Miron, D. Hrimiuc, H. Shimada and S. Sabau, G.S. Asanov, M.Crampin, P.L. Antonelli, S. Vacaru, S. Ikeda.

In the present chapter we will study the Finsler spaces considered asLagrange spaces and applying the mechanical principles. This methodsimplifies the theory of Finsler spaces. So, we will treat: Finsler met-ric, Cartan nonlinear connection derived from the canonical spray,Cartan metrical connection and its structure equations. Some exam-ples: Randers spaces, Kropina spaces and some new classes ofspacesmore general as Finsler spaces: the almost Finsler Lagrangespacesand the Ingarden spaces will close this chapter.

3.1 Finsler metrics

Definition 3.1.1.A Finsler space is a pairFn = (M,F(x,y)) whereMis a realn−dimensional differentiable manifold andF : TM → R is ascalar function which satisfies the following axioms:

47

48 3 Finsler Spaces

1F is a differentiable function onTM andF is continuous on the nullsection of the projectionπ : TM → M.

2F is a positive function.3F is positively 1-homogeneous with respect to the variablesyi

4The Hessian ofF2 with the elements:

gi j (x,y) =12

∂F2

∂yi∂y j (3.1.1)

is positively defined on the manifoldTM.Of course, the axiom 4 is equivalent with the following:

4′The pair(M,F2(x,y)) = LnF is a Lagrange space with positively de-

fined fundamental tensor,gi j . LnF will be called the Lagrange space

associated to the Finsler spaceFn. It follows that all properties ofthe Finsler spaceFn derived from the fundamental functionF2 andthe fundamental tensorgi j of the associated Lagrange spaceLn

F .

Remarks

1Sometimes we will ask forgi j to be of constant signature andrank(gi j (x,y)) = n on TM.

2Any Finsler spaceFn = (M,F(x,y)) is a Lagrange spaceLnF =

(M,F2(x,y)), but not conversely.

Examples

1A Riemannian manifold(M,γ i j (x)) determines a Finsler spaceFn = (M,F(x,y)), where

F(x,y) =√

γ i j (x)yiy j . (3.1.2)

The fundamental tensor isgi j (x,y) = γ i j (x).2Let us consider, in a preferential local system of coordinates, the

following function:

F(x,y) = 4√(y1)4+ ...+(yn)4. (3.1.3)

ThenF satisfy the axioms 1-4.

Remark 3.1.1.This example was given by B. Riemann.

3.1 Finsler metrics 49

3Antonelli-Shimada’s ecological metric is given, in a preferential lo-cal system of coordinates, onTM, by

F(x,y) = eφ L, φ = α ixi , (α i are positive constants),

and where

L = (y1)m+(y2)m+ · · ·+(yn)m1/m,m≥ 3, (3.1.4)

m being even.4Randers metric is defined by

F(x,y) = α(x,y)+β(x,y), (3.1.5)

whereα2(x,y) := ai j (x)y

iy j ,

(M,ai j (x)) being a Riemannian manifold and

β(x,y) := bi(x)yi .

The fundamental tensorgi j is expressed by:

gi j =α +β

αhi j +did j , hi j := ai j−

0l i

0l j ,

di = bi+0l i ,

0l i :=

∂α∂yi .

(3.1.5’)

One can prove thatgi j is positively defined under the conditionb2 =ai j bib j < 1. In this case the pairFn = (M,α +β ) is a Finsler space.

The first example motivates the following theorem:

Theorem 3.1.1.If the base manifold M is paracompact, then thereexist functions F: TM → R such that the pair(M,F) is a Finslerspaces.

Regarding the axioms 1−4 we can see without difficulties:

Theorem 3.1.2.The system of axioms of a Finsler space is minimal.

Some properties of Finsler spaceFn:

1The components of the fundamental tensorgi j (x,y) are 0-homo-geneous with respect toyi .

50 3 Finsler Spaces

2The components of 1-form

pi =12

∂F2

∂yi (3.1.6)

are 1-homogeneous with respect toyi .3The components of the Cartan tensor

Ci jk =14

∂ 3F2


∂gi j

∂yk (3.1.7)

are−1−homogeneous with respect toyi .

Consequently we have

Coi j = ysCsi j = 0. (3.1.8)

If Xi andYi are d-vector fields, then‖X‖2 := gi j (x,y)XiYi is a scalar field.< X,Y >:= gi j (x,y)XiY j is a scalar field.Assuming‖X‖u 6= 0, ‖Y‖u 6= 0 the angleϕ = ∠(X,Y) at a point

u∈ TM is given by

cosϕ =< X,Y > (u)‖X‖u · ‖Y‖u

.

The vectorsXu,Yu are orthogonal if〈X,Y〉(u) = 0.

Proposition 3.1.1.In a Finsler space Fn the following identities hold:

1F2(x,y) = gi j (x,y)yiy j

2 piyi = F2.

Proposition 3.1.2.1The 1-form

ω = pidxi (3.1.9)

is globally defined onTM.2The 2-form

θ = dω = dpi ∧dxi (3.1.10)

is globally defined onTM.3θ is a symplectic structure onTM.

3.2 Geodesics 51

Definition 3.1.2.A Finsler spaceFn = (M,F) is called reducible to aRiemann space if its fundamental tensorgi j (x,y) does not depend onthe variableyi .

Proposition 3.1.3.A Finsler space Fn is reducible to a Riemannspace if and only if the tensor Ci jk is vanishing onTM.

3.2 Geodesics

In a Finsler spaceFn = (M,F(x,y)) one can define the notion of arclength of a smooth curve.

Let c be a parametrized curve in the manifoldM:

c : t ∈ [0,1]→ (xi(t)) ∈U ⊂ M (3.2.1)

U being a domain of a local chart inM.The extensionc of c to TM has the equations

xi = xi(t), yi =dxi

dt(t), t ∈ [0,1]. (3.2.1’)

Thus the restriction of the fundamental functionF(x,y) to c is F(x(t),dxdt

(t)), t ∈ [0,1].

We define the “length” of curvec with extremitiesc(0),c(1) by thenumber

ℓ(c) =∫ 1

0F(x(t),

dxdt

(t))dt. (3.2.2)

The numberℓ(c) does not depend on a changing of coordinates onTM and, by means of 1-homogeneity of functionF, ℓ(c) does notdepend on the parametrization of the curvec. ℓ(c) depends on thecurvec, only.

We can choose a canonical parameter onc, considering the follow-ing functions= s(t), t ∈ [0,1]:

s(t) =∫ t

0F(x(τ),

dxdt

(τ))dτ.

This function is derivable and its derivative is

52 3 Finsler Spaces

dsdt

= F(x(t),dxdt

(t))> 0, t ∈ (0,1).

So the functions= s(t), t ∈ [0,1], is invertible. Lett = t(s), s∈ [s0,s1]be its inverse. The change of parametert → s has the property

F

(x(s),

dxds

(s)

)= 1. (3.2.3)

Variational problem on the functionalℓ will gives the curves onTM which extremize the arc length. These curves are geodesics of theFinsler spaceFn.

So, the solution curves of the Euler-Lagrange equations:

ddt

(∂F∂yi

)−

∂F∂xi = 0, yi =

dxi

dt(3.2.4)

are the geodesics of the spaceFn.

Definition 3.2.1.The curvesc = (xi(t)), t ∈ [0,1], solutions of theEuler-Lagrange equations (3.2.4) are called thegeodesicsof the FinslerspaceFn.

The system of differential equations (3.2.4) is equivalentto the fol-lowing system

ddt

∂F2

∂yi −∂F2

∂xi = 2dFdt

∂F∂yi , yi =

dxi

dt.

In the canonical parametrization, according to (3.2.3) we have:

Theorem 3.2.1.In the canonical parametrization the geodesics of theFinsler space Fn are given by the system of differential equations

Ei(F2) :=

dds

∂F2

∂yi −∂F2

∂xi = 0,yi =dxi

ds. (3.2.5)

Now, remarking thatF2 = gi j yiy j , the previous equations can bewritten in the form:

d2xi

ds2 + γ ijk

(x,

dxds

)dxj

dsdxk

ds= 0, yi =

dxi

ds, (3.2.6)

whereγ ijk are the Christoffel symbols of the fundamental tensorgi j :

3.3 Cartan nonlinear connection 53

γ ijk =

12

gir(

∂grk

∂x j +∂g jr

∂xk −∂g jk

∂xr

). (3.2.7)

A theorem of existence and uniqueness of the solution of differen-tial equations (3.2.6) can be formulated in the classical manner.

3.3 Cartan nonlinear connection

Considering the Lagrange spaceLnF =(M,F2) associated to the Finsler

spaceFn = (M,F) we can obtain some main geometrical object fieldof Fn.

So, Theorem 2.3.1, affirms:

Theorem 3.3.1.For the Finsler space Fn the equations

gi j E j(F2) := gi j

(ddt

∂F2

∂y j

)−

∂F2

∂x j = 0,yi =dxi

dt

can be written in the form

d2xi

dt2+2Gi

(x,

dxdt

)= 0,yi =

dxi

dt(3.3.1)

where2Gi(x,y) = γ i

jk(x,y)yjyk (3.3.1’)

Consequently the equations (3.3.1) give the integral curves of thesemispray:


∂∂yi . (3.3.2)

SinceGi are 2-homogeneous functions with respect toyi it followsthatS is aspray.

S determine a canonical nonlinear connectionN with the coeffi-cients

Nij =

∂Gi

∂y j =12

∂∂y j

γ irs(x,y)y

rys. (3.3.3)

N is called the Cartan nonlinear connection of the spaceFn.The tangent bundleT(TM), the horizontal distributionN and the

vertical distributionV give us the direct decomposition of vectorialspaces

54 3 Finsler Spaces

Tu(TM) = N(u)⊕V(u), ∀u∈ TM. (3.3.4)

The adapted basis toN andV is

(δ

δxi ,∂

∂yi

)and the dual adapted

basis is(dx,δyi), where

δδxi =

∂∂xi −N j

i (x,y)∂

∂y j

δyi = dyi +Nij(x,y)dxj .

(3.3.4’)

One obtains:

Theorem 3.3.2.1The horizontal curves in Fn are given by

xi = xi(t),δyi

dt= 0.

2The autoparallel curves of the Cartan nonlinear connectionN co-incide to the integral curves of the spray S,(3.3.2).

3.4 Cartan metrical connection

Let N(Nij) be the Cartan nonlinear connection of the Finsler space

Fn. According to section 5 of chapter 2 one introduces the canonicalmetricalN−linear connection of the spaceFn.

But, for these spaces the system of axioms, from Theorem 2.5.1 canbe given in the Matsumoto’s form [144], [145].

Theorem 3.4.1.1On the manifoldTM, for any Finsler space Fn =(M,F) there exists only one linear connection D, with the coeffi-cients CΓ = (Ni

j ,Fijk,C

ijk) which verifies the following axioms:

A1. The deflection tensor field Dij = yi| j vanishes.

A2. gi j |k = 0, (D is h−metrical).A3. gi j |k = 0, (D is v−metrical).A4. T i

jk = 0, (D is h−torsion free).

A5. Sijk = 0, (D is v−torsion free).

2The coefficients(Nij ,F

ijk,C

ijk) are as follows:

a. Nij are the coefficients (3.3.3) of the Cartan nonlinear connection.

3.4 Cartan metrical connection 55

b. Fijk,C

ijk are expressed by the generalized Christoffel symbols:

F ijk =

12

gis

(δgsk

δx j +δg js

δxk −δg jk

δxs

)

Cijk =

12

gis

(∂gsk

∂y j +∂g js

∂yk −∂g jk

∂ys

) (3.4.1)

3This connection depends only on the fundamental function F.

A proof can be find in the books [21], [164].The previous connection is named theCartan metrical connection

of the Finsler spaceFn.Now we can develop the geometry of Finsler spaces, exactly asthe

geometry of the associated Lagrange spaceLnF = (M,F2).

Also, in the case of Finsler space the geometrical modelH2n =

(TM,G,F) is an almost Kahlerian space.A very good example is provided by the Randers space, introduced

by R.S. Ingarden.A Randers space is the Finsler spaceFn = (M,α + β ) equipped

with Cartan nonlinear connectionN.It is denoted byRFn = (M,α +β ,N).The geometry of these spaces was much studied by many geome-

ters. A good monograph in this respect is the D. Bao, S.S. Chern andZ. Shen’s book [35].

The Randers spacesRFn can be generalized considering the FinslerspacesFn = (M,α + β ), whereα(x,y) is the fundamental functionof a Finsler spaceF ′n = (M,α). The Finsler spaceFn = (M,α +β )equipped with the Cartan nonlinear connectionN of the spaceF ′n =(M,α) is a generalized Randers space [152], [35]. Evidently, thisno-tion has some advantages, since we can take some remarkable FinslerspacesF ′n, (M. Anastasieiβ−transformation of a Finsler space [14]).

As an application of the previous notions, we define the notionof Ingarden spaceIF n, [115], [49]. This is the Finsler spaceFn =(M,α + β ) equipped with the nonlinear connectionN = γ i

jk(x)yk −

F ij (x), γ i

jk(x) being the Christoffel symbols of the Riemannian met-

ric ai j (x), which definesα2 = ai j (x)yiy j and the electromagnetic ten-sorF i

j (x) determined byβ = bi(x)yi . While the spacesRFn have not

the electromagnetic fieldF =12(Di j −D ji ), the Ingarden spaces have

such kind of tensor fields and they give us the remarkable Maxwellequations, [49]. Also, the autoparallel curve of the nonlinear connec-tion N are given by the known Lorentz equations.

56 3 Finsler Spaces

An example of a special Lagrange space derived from a Finslerone,[164] is as follows:

Let us consider the Lagrange spaceLn = (M,L(x,y)) with the fun-damental function

L(x,y) = F2(x,y)+β ,whereF is the fundamental function of a priori given Finsler spaceFn = (M,F) andβ = bi(x)yi .

These spaces have been calledthe almost Finsler Lagrange Spaces(shortly AFL-spaces), [164], [49]. They generalize the Lagrange spacefrom Electrodynamics.

Indeed, the Euler - Lagrange equations of AFL-spaces are exactlythe Lorentz equations

d2xi

dt2+ γ i

jk(x,y)dxj

dtdxk

dt=

12

F ij (x)

dxj

dt.

To the end of these three chapter we can do a general remark:The class of Riemann spacesRn is a subclass of the class of

Finsler spacesFn, the classFn is a subclass of class of LagrangespacesLn and this is a subclass of class of generalized LagrangespacesGLn. So, we have the following sequence of inclusions:

(I) Rn ⊂ Fn ⊂ Ln ⊂ GLn.

Therefore, we can say: The Lagrange geometry is the geometricstudy of the terms of the sequence of inclusions (I).

Chapter 4The Geometry of Cotangent Manifold

The geometrical theory of cotangent bundle(T∗M, π∗,M) on a real,finite dimensional manifoldM is important in differential geometry.Correlated with that of tangent bundle(TM,π,M), introduced in Ch.1, one obtains a framework for construction of Lagrangian and Hamil-tonian mechanical systems, as well as, for the duality between them,via Legendre transformation.

We study here the fundamental geometric objects onT∗M, as Liou-ville vector fieldC∗, Liouville 1-formω , symplectic structureθ =dω ,Poisson structure,N−linear connection etc.

4.1 Cotangent bundle

Let M be a realn−dimensional differentiable manifold. Its cotangentbundle(T∗M, π∗, M) can be constructed as the dual of the tangentbundle(TM,π ,M), [154]. If (xi) is a local coordinate system on adomainU of a chart onM, the induced system of coordinates onπ∗−1(U) is (xi , pi), (i, j,k, .. = 1,2, ...,n), p1, ..., pn are called “mo-mentum variables”. We denote(xi , pi) = (x, p) = u.

A change of coordinates onT∗M is given by

xi = xi(x1, ...,xn), rank

(∂ xi

∂x j

)= n

pi =∂x j

∂ xi p j .

(4.1.1)

57

58 4 The Geometry of Cotangent Manifold

The natural frame

(∂

∂xi ,∂

∂ pi

)= (∂i, ∂ i) is transformed by (4.1.1) in

the form

∂i =∂ x j

∂xi ∂ j +∂ p j

∂xi ∂ j , ∂ i =∂ xi

∂x j ∂ j . (4.1.2)

The natural coframe(dxi ,dpi) is changed by the rule

dxi =∂ xi

∂x j dxj ; dpi =∂x j

∂ xi dpj +∂ 2x j

∂ xi∂ xk p jdxk. (4.1.2’)

The Jacobian matrix of change of coordinates (4.1.1) is

J(u) =

∂ xi

∂x j 0

∂ p j

∂xi

∂x j

∂ xi

u

.

It followsdetJ(u) = 1 for everyu∈ T∗M.

So we get:

Theorem 4.1.1.The differentiable manifold T∗M is orientable.

One can prove that ifM is a paracompact manifold, thenT∗M isparacompact, too.

The kernel of differentialdπ∗ : TT∗M → T∗M is the vertical sub-bundleVT∗M of the tangent bundleTT∗M. The fibresVu of VT∗M,∀u∈ T∗M determine a distributionV on T∗M, called theverticaldis-tribution. It is locally generated by the tangent vector fields(∂ 1, ..., ∂ n).Consequently,V is an integrable distribution of local dimensionn.

Noticing the formulae (4.1.2), (4.1.2’) one can introduce the fol-lowing geometrical object fields:

C∗ = pi ∂ i , (4.1.3)

called the Liouville–Hamilton vector field onT∗M,

ω = pidxi , (4.1.4)

called the Liouville 1-form onT∗M,

θ = dω = dpi ∧dxi , (4.1.5)

4.1 Cotangent bundle 59

θ is a symplectic structure onT∗M. All these geometrical object fieldsdo not depend on the change of coordinates (4.1.1).

The Poisson brackets, on the manifoldT∗M are defined by

f ,g=∂ f∂ pi

∂g∂xi −

∂g∂ pi

∂ f∂xi , ∀ f ,g∈ F (T∗M). (4.1.6)

Of course, f ,g ∈ F (T∗M) and f ,g does not depend on thechange of coordinates (4.1.1).

Also, the following properties hold:

1 f ,g=−g, f,2 f ,g is R−linear in every argument,3 f ,g,h+g,h, f+h, f,g= 0 (Jacobi identity),4·,gh= ·,gh+·,hg.

The pairF (T∗M),, is a Lie algebra, called the Poisson–Lie al-gebra.

The relation between the structuresθ and , can be given bymeans of the notion of Hamiltonian system.

Definition 4.1.1.A differentiable Hamiltonian is a real functionH :T∗M → R which is ofC∞ class onTM∗ = T∗M \0 and continuouson T∗M.

An example: Letgi j (x) be aC∞−Riemannian structure onM. ThenH =

√gi j (x)pi p j is a differentiable Hamiltonian onT∗M.

Definition 4.1.2.A Hamiltonian system is a triple(T∗M,θ ,H).

Let us consider theF (T∗M)−modulesχ(T∗M) (tangent vectorfields onT∗M), χ∗(T∗M) (cotangent vector fields onT∗M).

The followingF (T∗M)−linear mapping

Sθ = χ(T∗M)→ χ∗(TM)

can be defined by

Sθ (X) = iXθ , ∀X ∈ χ(T∗M). (4.1.7)

One proves, without difficulties:

Proposition 4.1.1.Sθ is an isomorphism. We have:

Sθ

(∂

∂xi

)=−dpi , Sθ

(∂

∂ pi

)= dxi (4.1.6’)


Sθ (C∗) = ω. (4.1.7’)

Theorem 4.1.2.The following properties of the Hamiltonian system(T∗M,θ ,H) hold:

1There exists a unique vector field XH ∈ X (T∗M) for which:

iHθ =−dH. (4.1.8)

2The integral curves of the vector field XH are given by the Hamilton–Jacobi equations:

dxi

dt=

∂H∂ pi

,dpi

dt=−

∂H∂xi . (4.1.9)

Proof. 1The existence and uniqueness of the vector fieldXH is as-sured by the isomorphismSθ :

XH = S−1θ (−dH). (4.1.10)

XH is called theHamiltonian vectorfield.2The local expression ofXH is given (by (4.1.6)):

XH =∂H∂ pi

∂∂xi −

∂H∂xi

∂∂ pi

. (4.1.11)

Consequently: the integral curves of the vector fieldXH are givenby the Hamilton–Jacobi equations (4.1.8).

Along the integral curves ofXH , we have

dHdt

= H,H= 0.

Thus: The differentiable Hamiltonian H(x, p) is constant along theintegral curves of the Hamilton vector field XH.

The structuresθ and, have a fundamental property:

Theorem 4.1.3.The following formula holds:

f ,g= θ(Xf ,Xg), ∀ f ,g∈ X∗(T∗M),∀X ∈ X (T∗M). (4.1.12)

Proof. From (4.1.10):

f ,g= Xf g=−Xg f =−d f(Xg) = (iXf θ)(Xg) = θ(Xf ,Xg).

4.2 Variational problem. Hamilton–Jacobi equations 61

As a consequence, we obtain:

dxi

dt= H,xi,

dpi

dt= H, pi. (4.1.13)

It is remarkable the Jacobi method for integration of Hamilton–Jacobi equations (4.1.9). Namely, we look for a solution curvesγ(t)in T∗M of the form

xi = xi(t), pi =∂S∂xi (x(t)) (4.1.14)

whereS∈ F (M). Substituting in (4.1.9), we have

dxi

dt=

∂H∂ pi

(x(t))∂S∂xi (x(t));

dpi

dt=

∂ 2S∂xi∂x j

∂H∂ p j

=−∂H∂xi . (4.1.15)

It follows

dH

(x,

∂S∂x

)=

(∂H∂xi

∂H∂ pi

−∂H∂ pi

∂H∂xi

)dt = 0.

Consequently,H

(x,

∂S∂x

)=const, which is called the Hamilton–

Jacobi equation of Mechanics. This equation determines thefunctionSand the first equation (4.1.14) gives us the curvesγ(t).

The Jacobi method suggests to obtain the Hamilton–Jacobi equa-tion by the variational principle.

4.2 Variational problem. Hamilton–Jacobi equations

The variational problem for the Hamiltonian systems(T∗M,θ ,H) isdefined as follows:

Let us consider a smooth curvec defined on a local chartπ∗−1(U)of the cotangent manifoldT∗M by:

c : t ∈ [0,1]→ (x(t); p(t))∈ π∗−1(U)

analytically expressed by


xi = xi(t), pi = pi(t), t ∈ [0,1]. (4.2.1)

Consider a vector fieldV i(t) and a covector fieldη i(t) on the domainU of the local chart(U,ϕ) on M, and assume that we have

V i(0) =V i(1) = 0, η i(0) = η i(1) = 0

dVi

dt(0) =

dVi

dt(1) = 0.

(4.2.2)

The variations of the curvec determined by the pair(V i(t),η i(t))are defined by the curvesc(ε1,ε2):

xi = xi(t)+ ε1V i(t),

pi = pi(t)+ ε2η i(t), t ∈ [0,1],(4.2.3)

whenε1 andε2 are constants, small in absolute value such that theimage of any curvec belongs to the open setπ∗−1(U) in T∗M.

The integral of action of HamiltonianH(x, p) along the curvec isdefined by

I(c) =

1∫

0

[pi(t)

dxi

dt−H(x(t), p(t))

]dt. (4.2.4)

The integral of actionI(c(ε1,ε2)) is:

I(c(ε1,ε2)) =

∫ 1

0

[(pi + ε2η i(t))

(dxi

dt+ ε1

dVi

dt

)−

−H(x+ ε1V, p+ ε2η)]dt.

(4.2.5)

The necessary conditions in order thatI(c) is an extremal value ofI(c(ε1,ε2)) are

∂ I(c(ε1,ε2))

∂ε1

∣∣∣∣ε1=ε2=0

= 0,∂ I(c(ε1,ε2))

∂ε2

∣∣∣∣ε1=ε2=0

= 0. (4.2.6)

4.2 Variational problem. Hamilton–Jacobi equations 63

Under our conditions of differentiability, the operators∂

∂ε1,

∂∂ε2

and the operator of integration commute. Therefore, from (4.2.5) wededuce: ∫ 1

0

[pi(t)

dVi

dt(t)−

∂H∂xi V

i

]dt = 0

∫ 1

0

(dxi

dt−

∂H∂ pi

)η idt = 0.

(4.2.7)

Denoting:Ei(H) =

dpi

dt+

∂H∂xi (4.2.8)

and noticing the conditions (4.2.2) one obtain that the equations(4.2.7) are equivalent to:

1∫

0

Ei(H)V idt = 0;

1∫

0

[dxi

dt−

∂H∂ pi

]η idt = 0. (4.2.9)

But (V i ,η i) are arbitrary. Thus, (4.2.9) lead to the following result:

Theorem 4.2.1.In order to the integral of action I(c) to be an ex-tremal value for the functionals I(c) is necessary that the curve c tosatisfy the Hamilton–Jacobi equations:

dxi

dt=

∂H∂ pi

,dpi

dt=−

∂H∂xi . (4.2.10)

The operatorEi(H) has a geometrical meaning:

Theorem 4.2.2.Ei(H) is a covector field.

Proof. With respect to a change of local coordinates(4.1.1) on themanifoldT∗M, we have

∫ 1

0

Ei(H)V idt−

∫ 1

0

Ei(H)V idt =

=

∫ 1

0

[Ei(H)

∂ xi

∂x j −E j(H)

]V jdt = 0.


Since the vectorV i is arbitrary, we obtain:

Ei(H)

∂ xi

∂x j =E j(H).⊓⊔

A consequence of the last theorem: The Hamilton–Jacobi equationhave a geometrical meaning on the cotangent manifoldT∗M.

The notion of homogeneity for the Hamiltonian systems is definedin the classical manner [174].

A smooth functionf : T∗M → R is calledr−homogeneousr ∈ Zwith respect to the momenta variablespi if we have:

LC∗ f = C∗ f = p : ∂ i f = r f . (4.2.11)

A vector fieldX ∈ χ(T∗M) is r−homogeneous if

LC∗X = (r −1)X, (4.2.12)

whereLC∗X = [C∗;X]. So, we have

1∂

∂xi ,∂

∂ pi= ∂ i are 1- and 0-homogeneous.

2 If f ∈ F (T∗M) is s−homogeneous andX ∈ χ(T∗M) is r−homo-geneous thenf X is r +s−homogeneous.

3C∗ = pi ∂ i is 1-homogeneous.4 If X is r−homogeneous andf is s−homogeneous thenX f is r +

s−1−homogeneous.5 If f is s−homogeneous, then∂ i ∂ f is s−1−homogeneous.

Analogously, forq−forms ω ∈ Λq(T∗M). The q−form ω is r−ho-mogeneous if

LC∗ω = rω. (4.2.13)

It follows:

1′ If ω ,ω ′ ares− respectivelys′−homogeneous, thenω ∧ω ′ is s+s′−homogeneous.

2′ dxi ,dpi are 0− respectively 1−homogeneous.3′ The Liouville 1−form ω = pidxi is 1-homogeneous.4′ The canonical symplectic structureθ is 1-homogeneous.


4.3 Nonlinear connections

On the manifoldT∗M there exists a remarkable distribution. It isthe vertical distributionV. As we know,V is integrable, having(∂ 1, ∂ 2, ..., ∂ n) as a local adapted basis and dimV = n=dimM.

Definition 4.3.1.A nonlinear connectionN on the manifoldT∗M isa differentiable distributionN on T∗M supplementary to the verticaldistributionV:

TuT∗M = Nu⊕Vu, ∀u∈ T∗M. (4.3.1)

N is called thehorizontaldistribution onT∗M.

It follows that dimN = n.When a nonlinear connectionN is given we can consider anadapted

basis

(δ

δx1 , ...,δ

δxn

)expressed in the form

δδxi =

∂∂xi +Ni j

∂∂ p j

(i = 1,2, ...,n). (4.3.2)

The system of functionNi j (x, p) is well determined by (4.3.1). Itis called system of coefficients of the nonlinear connectionN. Thissystem defines a geometrical object field onT∗M.

The set of vector fields

(δ

δxi , ∂i

)give us an adapted basis to the

direct decomposition (4.3.1). Its dual adapted basis is(dxi ,δ pi), (i =1, ...,n), where

δ pi = dpi −Nji dxj . (4.3.3)

It is not difficult to prove that ifM is a paracompact manifold, thenon the cotangent manifoldT∗M there exists a nonlinear connection.

Let N be a nonlinear connection with the coefficientsNi j (x, p) anddefine the set of functionτ i j (x,y) by

τ i j =12(Ni j −Nji). (4.3.4)

It is not difficult to see thatτ i j is transformed, with respect to (4.1.1)as a covariant tensor field on the base manifoldM. So, it is a distin-guished tensor field, shortly ad−tensor oftorsion of the nonlinearconnection.τ i j = 0 has a geometrical meaning. In this caseN is asymmetric nonlinear connection.


With respect to a symmetric nonlinear connectionN the symplecticstructureθ can be written in an invariant form:

θ = δ pi ∧dxi , (4.3.5)

and the Poisson structure, can be expressed in the invariant form:

f ,g=∂ f∂ pi

δgδxi −

∂g∂ pi

δ fδxi . (4.3.6)

Of course, we can consider the curvature tensor ofN as thed−tensorfield

Ri jh =δNji

δxh −δNhi

δx j . (4.3.7)

It is given by [δ

δx j ,δ

δxh

]= Ri jh ∂ i . (4.3.8)

ThereforeRi jh(x, p) is the integrability tensor of the horizontal dis-tribution N. ThusN is an integrable distribution if and only if thecurvature tensorRi jh(x, p) vanishes.

A curvec : I ⊂ R→ c(t) ∈ T∗M. Imc⊂ π∗(U) expressed by (xi =

xi(t), pi = pi(t), t ∈ I ). The tangent vectordcdt

can be written in the

adapted basis as:dcdt

=dxi

dtδ

δxi +δ pi

dt∂

∂ pi,

whereδ pi

dt=

dpi

dt−Nji (x(t), p(t))

dxj

dt.

The curvec is horizontal ifδ pi

dt= 0. We obtain the system of differ-

ential equations which characterize the horizontal curves:

xi = xi(t),dpi

dt−Nji (x, p)

dxj

dt= 0. (4.3.9)

Let gi j (x,y) be ad−tensor by (d−means distinguished) with thepropertiesgi j = g ji and det(gi j ) 6= 0 onT∗M. Its contravariantgi j (x,y)

can be considered,gi j gts = δ si . As usually, we put

δδxi = δ i ,

∂∂ pi

= ∂


and ∂i = gi j ∂ j . So, one can consider the followingF (T∗M)−linearmappingF : X (T∗M)→ X (T∗M):

F(δ i) =−∂i , F(∂i) = δ i , (i = 1, ...,n). (4.3.10)

It is not difficult to prove, [174]:

Theorem 4.3.1.1The mappingF is globally defined on T∗M.2 F is a tensor field of type(1,1) on T∗M.3The local expression ofF in the adapted basis(δ i, ∂ i) is

F=−gi j ∂ i ⊗dxj +gi j δ i ⊗δ p j . (4.3.11)

4 F is an almost complex structure on T∗M determined by N and bygi j (x, p), i.e.

F F=−I . (4.3.12)

Also, nonlinear symmetric connectionN being considered, we candefine the tensor:

G(x, p) = gi j (x, p)dxi ⊗dxj +gi j δ pi ⊗δ p j . (4.3.13)

If d−tensorgi j (x, p) has a constant signature onT∗M (for instanceit is positively defined), then it follows:

Theorem 4.3.2.1G is a Riemannian structure on T∗M determinedby N and gi j .

2The distributions N and V are orthogonal with respect toG.3The pair(G, F) is an almost Hermitian structure on T∗M.4The associated almost symplectic structure to(G, F) is the canoni-

cal symplectic structureθ = δ pi ∧dxi .

Remarking that the vector fields(δ i , ∂ i) are transformed by (4.1.1)in the form

δ i =∂x j

∂ xi δ j ,˜∂

i=

∂ xi

∂x j ∂ j , (4.3.14)

and the 1-forms(dxi ,δ pi) are transformed by

dxi =∂ xi

∂x j dxj , δ pi =∂x j

∂ xi δ p j , (4.3.13’)

we can consider the horizontal and vertical projectors withrespect tothe direct decomposition (4.3.1):


h=δ

δxi ⊗dxi , v= ∂ i ⊗δ pi . (4.3.15)

They have the properties:

h+v= I , h2 = h, v2 = v, hv+vh= 0.

We sethX = XH , vX = XV , ∀X ∈ X (T∗M)

ωH = ω h, ωv = ω v, ∀ω ∈ X ∗(T∗M).

Therefore, for every vector fieldX on T∗M, represented in adaptedbasis in the form

X = Xiδ i + Xi ∂ i

we have

XH = hX = Xi δδxi , XV = vX = Xi ∂ i

where the coefficientsXi(x, p) andXi(x, p) are transformed by (4.1.1):

Xi =∂ xi

∂x j Xj , ˜Xi =

∂x j

∂ xi Xj .

For this reason,Xi(x, p) are the coefficients of a distinguished vectorfield andXi(x, p) are the coefficients of a distinguished covector field,shortly denoted byd−vector (covector) fields.

Analogously, for the 1-formω :

ω = ω idxi + ω iδ pi .

Therefore,ωH = ω idxi , ωV = ω iδ pi .

Thenω i(x, p) are component of an oned−form onT∗M andω i(x, p)are the components of ad−vector field onT∗M.

On the same way, we can define a distinguished(d−) tensor field.A d−tensor field can be represented in adapted basis in the form

T = T i1...irj1... js

(x, p)δ i1 ⊗ ...⊗ ∂ js⊗dxj1 ⊗ ...⊗δ pr . (4.3.16)

Its coefficients are transformed by (4.1.1) in the form:

4.4 N−linear connections 69

T i1...irj1... js

(x, p) =∂ xi1

∂xh1...

∂ xir

∂xhr

∂xk1

∂ x j1...

∂xk2

∂ x jsTh1...hr

k1...ks(x, p). (4.3.15’)

So, ad−tensor fieldT can be given by the coefficientsT i1...irj1... js

(x, p)whose rule of transformation, with respect to (4.1.1) is thesame withthat of a tensor of the same type on the base manifoldM of cotangentbundle(T∗M,M,π∗).

One can speak of thed−tensor algebra onT∗M, which is not diffi-cult to be defined.

4.4 N−linear connections

As we know, a nonlinear connectionN determines a direct decompo-sition (4.3.1) in respect to which we have

X = XH +XV , ω = ωH +ωV , ∀X ∈ X (T∗M), ∀ω ∈ X∗(T∗M).

(4.4.1)Assuming thatN is a symmetric nonlinear connection we can give:

Definition 4.4.1.A linear connectionD on the manifoldT∗M is calledanN−linear connection if:

1D preserves by parallelism the distributionsN andV.2The canonical symplectic structureθ = δ pi ∧dxi has the associate

tensorθ = δ pi ⊗dxi parallel with respect toD:

Dθ = 0. (4.4.2)

It follows that:DXh= DXv= 0

DXY = DXHY+DXVY.(4.4.3)

We obtain new operators of derivations in the algebras ofd−tensors,defined by

DHX = DXH , DV

X = DXV , ∀X ∈ X (T∗M). (4.4.4)

We have


DX = DHX +DV

X; DHX f = XF f , DV

X f = XVX

DHX ( fY) = XH fY+ f DH

XY; DVX( fY) = XV fY+ f DV

XY

DHf X = f DH

X , DVf X = f DV

X

DHX θ = 0, DV

Xθ = 0.

(4.4.5)

The operatorsDH ,DV have the property of localization. They havethe similarly properties with covariant derivative but they are notcovariant derivative.DH is calledh−covariant derivative andDV iscalledv−covariant derivative. They act on the 1-formsω on T∗M bythe rules:

(DHX ω)(Y) = XHω(Y)−ω(DH

XY)

(DVXω)(Y) = XVω(Y)−ω(DV

XY).(4.4.6)

The extension of these operators to thed−tensor fields is immediate.The torsionT of anN−linear connection has the form

Π(X,Y) = DXY−DYX− [X,Y]. (4.4.7)

It is characterized by the following vector fields:

T(XH ,YH), T(XH ,YV), T(XV ,XV).

Taking theh− andv−components we obtain thed−tensor of torsion

T(XH,YH) = hT(XH ,YH)+vT(XH,YH), etc.

Proposition 4.4.2.The d−tensors of torsion of an N−linear connec-tion D are:

hT(XH,YH) = DHXYH −DH

Y XH − [XH ,YH ]H

hT(XH,YV) =−DVYXH +[XH ,YV ]H

vT(XH ,YH) =−[XH ,YH ]V

vT(XH ,YV) = DHXYV − [XH ,YV ]V

vT(XV ,YV) = DVXYV −DV

YXV − [XV ,YV ]V = 0.

(4.4.8)

The curvatureR of anN−linear connectionD is given by


R(X,Y)Z = (DXDY −DYDX)Z−D[X,Y]Z. (4.4.9)

Remarking that the vector fieldR(X,Y)ZH is horizontal one andR(X,Y)ZV is a vertical one, we have

v(R(X,Y)ZH) = 0, h(R(X,Y)ZV) = 0. (4.4.10)

We will see that thed−tensors of curvature ofD are:

R(XH,YH)ZH , R(XH,YV)ZH , R(XV,YV)ZH . (4.4.11)

By means of (4.4.9) one obtains:Proposition 4.4.3.1TheRicci identities of D are

[DX,DY]Z = R(X,Y)Z−D[X,Y]Z. (4.4.12)

2TheBianchi identities are given by

∑(XYZ)

(DXT)(Y,Z)−R(X,Y)Z+T(T(X,Y),Z)= 0,

∑(X,Y,Z)

(DXR)(U,Y,Z)−R(T(X,Y),Z)U= 0,(4.4.13)

where ∑(XYZ)

means the cyclic sum.

The previous formulae can be expressed in local coordinates, takingδ i, ∂ i , as the vectorsX,Y,Z,U . But, first of all, we must introduce thelocal coefficients of anN−linear connectionD. In the adapted basis(δ i , ∂ i) we take into account the properties:

Dδ j= DH

δ j, D∂ j = DV

∂ j . (4.4.14)

Then, the following theorem holds:

Theorem 4.4.1.1An N−linear connection D on T∗M can be uniquelyrepresented in adapted basis(δ i , ∂ i) in the form:

Dδ jδ i = Hh

i j δ h, Dδ j∂ i =−H i

h j∂h,

D∂ j δ i =Ch ji δ h, D∂ j ∂ i =−Ci j

h ∂ h.(4.4.15)


2With respect to (4.4.1) on T∗M the coefficients Hijk(x, p) transformby the rule

H irs

∂xr

∂ x j

∂xs

∂ xk =∂ xi

∂xr Hrjk −

∂ 2xi

∂x j∂xk (4.4.16)

and Cjki (x, p) is a d−tensor of type(2,1).

The proof of this theorem is not difficult.Conversely, ifN is an a priori given nonlinear connection and a

set of functionsH ijk(x, p),C

jki (x, p) on T∗M, verifying 2 is given,

then there exists an uniqueN−linear connectionD with the property(4.4.16).

The action ofD on the adapted cobasis(dxi ,δ pi) is given by

Dδ jdxi =−H i

k jdxk, Dδ jδ pi = Hk

i j δ pk,

D∂ j dxi =−Ci jk dxk, D∂ j δ pi =Ck j

i δ pk.(4.4.17)

The pairDΓ (N) = (H ijk,C

jki ) is called the system of coefficients ofD.

Let us consider ad−tensor fieldT with the local coefficientsT i,...,ir

j1,..., js(x, p), (see (4.3.15)) and a horizontal vector fieldX = XH =

Xiδ i .By means of previous theorem we obtain forh−covariant deriva-

tion DHX of tensorT:

DHX T = XkT i1...ir

j1... js,|kδ i1 ⊗ ...⊗ ∂ js⊗dxj1 ⊗ ...⊗δ pir , (4.4.18)

where

T i1...irj1... js|k

= δ kTi1...irj1... js

+Tmi2...irj1... js

H i1mk+ ...+T ir1...m

j1... jsH ir

mk−

−T i1...irm... js

Hmj1k− ...−T i1...ir

j1...mHm

jsk.

(4.4.16’)

The operator “|” is called h−covariant derivative with respect toDΓ (N) = (H i

jk,Cjki ).

Now, takingX = XV = Xi ∂ i , thev−covariant derivativeDVXT has

the following form

DVXT = Xk T i1...ir

j1... js

∣∣∣k

δ i ⊗ ...⊗ ∂ js⊗dxj1 ⊗ ...⊗δ pir , (4.4.19)


where

T i1...irj1... js

∣∣∣k= ∂ kT i1...ir

j1... js+Tm...ir

j1... jsCi1k

m + ...+T i1...mj1... js

Cirkm −

−T i1...irm... js

Cmkj1

− ...−T i1...irj1...m

Cmkjs .

(4.4.17’)

The operator “—” is called thev−covariant derivative.

Proposition 4.4.4.The following properties hold:

1T i1...irj1... js|k

is a d−tensor of type(r,s+1).

2 T i1...irj1... js

∣∣∣k

is a d−tensor of type(r +1,s).

3 f|m =δ fδxm, f |m = ∂ m f .

4 Xi|k = δ kXi +XmH i

mk; Xi|k = ∂ kXi +XmCikm,

ω i|k = δ kω i −ωmHmik ; ω1|

k = ∂ kω i −ωmCmki .

(4.4.20)

5The operators “|” and “ |” are distributive with respect to addi-tion and verify the rule of Leibnitz with respect to tensor product ofd−tensors.

Let us consider thedeflectiontensors ofDΓ (N):

∆i j = pi| j , δji = pi |

j . (4.4.21)

One gets

∆i j = Ni j − pmHmi j ;

−δ ij = δ i

j − pmCm ji . (4.4.19’)

Proposition 4.4.5.If H ijk(x, p) is a system of differentiable functions,

verifying(4.4.16), then Ni j (x, p) given by

Ni j = pmHmi j (4.4.22)

determine a nonlinear connection in T∗M.

As in the case of tangent bundle, one can prove that if the basemanifold M is paracompact, then onT∗M there exist theN−linearconnections.


Indeed, onM there exists a Riemannian metricgi j (x). Then the pairDΓ (N) = (γ i

jk,0) is an N−linear connection onT∗M, γ ijk(x) being

the Christoffel coefficients andNi j = pmγmi j , are the coefficients of a

nonlinear connection onT∗M.In the adapted basis(δ i , ∂ i), the Ricci identitites (4.4.12) can be

written in the form

Xi| j |h−Xi

|h| j = XmR im jh−Xi

|mTmjh −Xi |mRm jh

Xi| j |

h−Xi|h| j = XmP i hm j −Xi

|mCmhj −Xi|mPh

m j

Xi| j |h−Xi |h| j = XmS i jhm −Xi|mS jh

m ,

(4.4.23)

where

T ijh = H i

jh−H ih j, S jh

i =C jhi −Ch j

i , Pijh = H i

jh− ∂ iNh j (4.4.24)

andRm jh,Cmhi are thed−tensors of torsion andR i

k jh, P i hk j , S i jh

k arethed−tensors of curvature:

R ik jh = δ hH i

k j −δ jH ikh+Hm

k jHimh−Hm

khHikhH

im j−C im

k Rm jh,

P i hk j = ∂ hH i

k j −Cihk | j +Cim

k Phm j,S

i jhk = ∂ hCi j

k − ∂ jCihk +

+C m jk C ih

m −C mhk C i j

m .(4.4.25)

Evidently, thesed−tensors of curvature verify:

R(δ jδ h)δ k = R ik h jδ i ; R(δ j , ∂ h)δ k = P h i

k j δ i ,

R(∂ j , ∂ h)δ k = Sih jk δ i ,

(4.4.26)

and

R(δ jδ h)∂ k = R ki h j ∂

i ; R(δ j , ∂ h)∂ k =−P k hi j ∂ i ,

R(∂ j , ∂ h)∂ k =−Shk ji ∂ j .

(4.4.23’)

The Bianchi identities (4.4.13), in adapted basis(δ i , ∂ i) can be writ-ten without difficulties.


Applications:

1For ad− tensorgi j (x, p) the Ricci identities are

gi j|k|h−gi j

|h|k=gm jR im kh+gimR j

mkh−gi j|mTm

kh−gi j |mRmkh,

gi j|k|

h−gi j |h|k=gm jP i hm k+gimP j h

m k−gi j|mC mh

k −gi j |mPhmk,

gi j |k|h−gi j |h|k = gm jS ikhm +gimS jkh

m −gi j |mSkhm .

(4.4.27)

In particular, ifgi j is covariant constant with respect toN−connectionD, i.e.

gi j|k = 0, gi j |k = 0, (4.4.28)

then, from (4.4.27) we have:

gm jR im kh+gimR j

m kh= 0,gm jP i h

m k+gimP j hm k = 0,

gm jS ikhm +gimS jkh

m = 0.

(4.4.25’)

Such kind of equations will be used for theN−linear connectionscompatible with a metric structureG in (4.3.13).

The Ricci identities applied to the Liouville–Hamilton vector fieldC∗ = pi ∂ i lead to the important identities, which imply the deflectiontensors∆i , and−δ i

j .

Theorem 4.4.2.Any N−linear connection D on T∗M satisfies the fol-lowing identities:

∆i j |k−∆ik| j =−pmR mi jk −∆imTm

jk −−δ m

i Rm jk,

∆i j |k−−δ k

i | j =−pmPm ki j −∆imCmk

j −−δ mi Pi

m j,−δ j

i |k−−δ k

j |j =−pmSm jk

i −−δ mi S jk

m .

(4.4.29)

One says that areN−linear connectionD is of Cartan type if∆i j =

0, −δ ij = δ i

j .

Proposition 4.4.6.Any N−linear connection D of Cartan type satis-fies the identities:

pmRmi jk +Ri jk = 0, pmP m k

i j +Pki j = 0, pmSm jk

i +S jki = 0. (4.4.30)


Finally, we remark that we can explicitly write, in adapted basis,the Bianchi identities (4.4.13).

4.5 Parallelism, paths and structure equations

Consider the Hamiltonian systems(T∗M,H,θ), anN−linear connec-tion DΓ (N) = (H i

jk,Cjk

i ).A curvec : I → T∗M, locally represented by

xi = xi(t), pi = pi(t), (4.5.1)

has the tangent vectordcdt

in adapted basis(δ i , ∂ i):

dcdt

=dxi

dtδ i +

δ pi

dt∂ i , (4.5.2)

withδ pi

dt=

dpi

dt− Nji

dxj

dt. The operator of covariant derivativeD

along the curvec is

Dc = DHc +DV

c =dxi

dtDδ i

+δ pi

dtD∂ i .

If X = Xiδ i + Xi ∂ i is a vector field then the operatorX of covariantdifferential acts on the vector fieldsD by the rule:

DX = (dxi +Xmω im)δ i +(dXi − Xmωm

i )∂i , (4.5.3)

whereω i

j = H ijkdxk+Cik

j δ pk (4.5.4)

is calledthe connection1−form.

Therefore, the covariant differentialDXdt

is:

DXdt

=

(dXi

dt+Xmω i

m

dt

)δ i +

(dXi

dt− Xm

ωmi

dt

)∂ i , (4.5.5)

4.5 Parallelism, paths and structure equations 77

wheredXi

dtis

dXi

dt(x(t), p(t)), t ∈ I and

ω ij

dt= H i

jk(x(t), p(t))dxk

dt+C ik

j (x(t), p(t))δ pk

dt. (4.5.6)

As usually, we introduce:

Definition 4.5.1.The vector fieldX = Xi(x, p)δ i + Xm(x, p)∂ i is par-allel, with respect toDΓ (N), along smooth curvec : I → T∗M ifDXdt

= 0,∀t ∈ I .

From (4.5.5) one obtains:

Theorem 4.5.1.The vector field X= Xiδ i + Xi ∂ i is parallel, with re-spect to the N−linear connection DΓ (H i

jk,Cjki ), along of curve c if,

and only if, the functions Xi , Xi, (i = 1, ...,n) are solutions of the dif-ferential equations:

dXi

dt+Xmω i

m

dt= 0,

dXi

dt− Xm

ωmi

dt= 0. (4.5.7)

The proof is immediate, by means of (4.5.5).A theorem of existence and uniqueness for the parallel vector fields

along a given parametrized curvec in the manifoldT∗M can be for-mulated in the classical manner. Let us consider the case when a vec-tor field X is absolute parallel onT∗M, with respect to anN−linearconnectionD, i.e.DX = 0 onT∗M, [174].

Using the formula (4.5.3),DX = 0 if and only if

dXi +Xmω im = 0, dXi − Xmωm

i = 0.

Remarking (4.5.4), and that we have

dXi = Xi|kdxk+Xi|kδ pk

dXi = Xi |kdxk+ Xi|kδ pk

it follows that the equationDX = 0 along any curvec on T∗M isequivalent to

Xi| j = 0, Xi| j = 0

Xi| j = 0, Xi|j = 0.

(4.5.8)


The differential consequence of the previous system are given by theRicci identities (4.4.26), taken modulo (4.5.8). They are

XhR ih jk = 0, XhP i k

h j = 0, XhS i jkh = 0

XhR hi jk = 0, XhP h k

i j = 0, XhSh jki = 0.

(4.5.9)

But, Xi, Xi being arbitrary, it follows

Theorem 4.5.2.The N−linear connection DΓ (N) is with the abso-lute parallelism of vectors if, and only if, the curvature tensorR of Dvanishes.

Definition 4.5.2.The curvec : I → T∗M is called autoparallel forN−linear connectionD if Dc · c= 0.

Taking into account the fact that ˙c=dcdt

=dxi

dtδ i +

δ pi

dt∂ i , one ob-

tains

Theorem 4.5.3.A curve c: I → T∗M is autoparallel with respect toDΓ (N) if, and only if, the functions xi(t), pi(t), t ∈ I, are solutions ofthe system of differential equations

d2xi

dt2+

dxs

dtω i

s

dt= 0,

ddt

δ pi

dt−

δ ps

dt

ωsi

dt= 0. (4.5.10)

Of course, a theorem of existence and uniqueness for the autoparallelcurves can be formulated.

Definition 4.5.3.An horizontal path ofDΓ (N) is an horizontal au-toparallel curve.

Theorem 4.5.4.The horizontal paths of DΓ (N) are characterized bythe differential equations

d2xi

dt2+H i

jk(x, p)dxj

dtdxk

dt= 0. (4.5.11)

The curvec : I → T∗M is vertical at the pointxi0 = xi(t0), t0 ∈ I of

M if c(t0) ∈V (vertical distribution).A vertical pathat the point(xi

0) ∈ M is a vertical autoparallel curveat point(xi

0).

4.5 Parallelism, paths and structure equations 79

Theorem 4.5.5.The vertical paths at a point x0 = (xi0) ∈ M with re-

spect to DΓ (N) are solutions of the differential equations

d2pi

dt2−C jk

i (x0, p)dpj

dtdpk

dt= 0, xi = xi

0. (4.5.12)

Now, we can write the structure equations ofDΓ (N) = (H ijk,C

jki ),

remarking that the following geometric objects are ad−vector fieldand ad−covector field, respectively:

d(dxi)−dxm∧ω im; dδ pi +δ pm∧ωm

i

anddω ij −ωm

j ∧ω im is ad−tensor of type (1,1). Hered is the operator

of exterior differential.By a straightforward calculus we can prove:

Theorem 4.5.6.For any N−linear connection D with the coefficientsDΓ (N) = (H i

jk,Cjki ) the following structure equations hold:

d(dxi)−dxm∧ω im =−Ω i

d(δ pi)+δ pm∧ωmi =−Ωi

dω ij −ωm

j ∧ω im =−Ω i

j ,

(4.5.13)

whereΩ i ,Ωi are2−forms of torsion:

Ω i =12

T ijkdxj ∧dxk+Cik

j dxj ∧δ pk,

Ωi =12

Ri jkdxj ∧dxk+Pki j dxj ∧δ pk+

12

S jki δ p j ∧δ pk,

(4.5.14)

and whenΩ ij is 2-forms of curvature:

Ω ij =

12

R ij kmdxk∧dxm+P i m

j k dxk∧δ pm+12

S jkmi δ pk∧δ pm. (4.5.15)

Remarks.1The torsion and curvatured−tensor forms (4.5.14) and (4.5.15) are

expressed in Section 4 of this chapter.


2The Bianchi identities can be obtained exterior differentiating thestructure equations modulo the same equations.

Chapter 5Hamilton spaces

The notion of Hamilton space was defined and investigated by R.Miron in the papers [154], [174], [175]. It was studied by D. Hrim-iuc [109], H. Shimada [223] et al. It was applied by P.L. Antonelli,S. Vacaru, D. Bao et al. [251]. The geometry of these spaces isthegeometry of a Hamiltonian system(T∗M,θ ,H), whereH(x, p) is afundamental function of an Hamilton spaceHn = (M,H(x, p)). Con-sequently, we can apply the geometric theory of cotangent manifoldT∗M presented in the previous chapter, establishing the all fundamen-tal geometric objects of the spacesHn.

As we see, the geometry of Hamilton spaces can be studied as dualgeometry, via Legendre transformation of the geometry of Lagrangespaces.

In this chapter we study the geometry of Hamilton spaces, com-bining these two methods. So, we defined the notion of HamiltonspaceHn, the canonical non-linear connectionN of Hn, the metri-cal structure and the canonical metrical linear connectionof Hn. Thefundamental equations ofHn are the Hamilton–Jacobi equations. Thenotion of parallelism with respect to aN−metrical connection andits consequences are studied. As applications we study the notion ofHamilton spaces of the electrodynamics. Also the almost Kahlerianmodel of the spacesHn is pointed out.

5.1 Notion of Hamilton space

Let M be a real differential manifold of dimensionn and(T∗M,π∗,M)its cotangent bundle. A differentiable Hamiltonian is a mapping H :T∗M → R differentiable onT∗M = T∗M \0 and continuous on thenull section.

81

82 5 Hamilton spaces

The Hessian ofH, with respect to the momenta variablepi of dif-ferential HamiltonianH(x, p) has the components

gi j =12

∂ 2H∂ pi∂ p j

. (5.1.1)

Of course, the Latin indicesi, j, ... run on the set1, ...,n.The set of functionsgi j (x, p) determine a symmetric contravariant

of order 2 tensor field onTM∗. H(x, p) is called regular if

det(gi j (x, p)) 6= 0 onT∗M. (5.1.2)

Now, we can give

Definition 5.1.1.A Hamilton space is a pairHn = (M,H(x, p)) whereM is a smooth real,n−dimensional manifold, andH is a function onT∗M having the properties

1H : (x, p)∈ T∗M →H(x, p)∈R is a differentiable function onT∗Mand it is continuous on the null section of the natural projectionπ∗T∗M → M.

2The Hessian ofH with respect to momentapi given by‖gi j (x, p)‖,(5.1.1) is nondegenerate i.e. (5.1.2) is verified onT∗M.

3Thed−tensor fieldgi j (x, p) has constant signature on the manifoldT∗M.

One can say that the Hamilton spaceHn = (M,H(x, p)) has as fun-damental function a differentiable regular Hamiltonian for which itsfundamentalor metrictensorgi j (x, p) is nonsingular and has constantsignature.Examples

1 If Rn = (M,gi j (x)) is a Riemannian space thenHn = (M,H(x, p))with

H(x, p) =1

mcgi j (x)pi p j (5.1.3)

is an Hamilton space.2The Hamilton space of electrodynamics is defined by the funda-

mental function

H(x, p) =1

mcgi j (x)pi p j −

2emc2Ai(x)pi −

e2

mc2Ai(x)Ai(x), (5.1.4)

5.1 Notion of Hamilton space 83

wherem,c,e are the known physical constants,gi j (x) is a pseudo-Riemannian tensor andAi(x) are ad−covector (electromagneticpotentials) andAi(x) = gi j A j .

Remark 5.1.1.The kinetic energy for a Riemannian metricRn =(M,gi j (x)) is given by

T(x, x) =12

gi j (x)xi x j

and for the Hamilton spaceHn=(M,H(x, p))with H(x, p)=gi j (x)pi p jit is

T∗(x, p) =12

gi j pi p j .

Therefore, generally, for an Hamilton spaceHn = (M,H(x, p)) isconvenient to introduce the energy given by the Hamiltonian

H (x, p) =12

H(x, p). (5.1.5)

One obtain:∂ i ∂ j

H = gi j (x, p). (5.1.6)

Consider the Hamiltonian system(T∗M,θ ,H ), whereθ is the nat-ural symplectic structure onT∗M,θ = dpi ∧dxi (Ch. 4). The isomor-phism Sθ , defined by (4.1.7) can be used and Theorem 4.2 can beapplied.

One obtainTheorem 5.1.1.For any Hamilton space Hn = (M,H(x, p)) the fol-lowing properties hold:

1There exists a unique vector field XH ∈ X (T∗M) for which

iHθ =−dH . (5.1.7)

2XH is expressed by

XH =∂H

∂ pi

∂∂xi −

∂H

∂xi

∂∂ pi

. (5.1.8)

3The integral curves of XH are given by the Hamilton–Jacobi equa-tions

dxi

dt=

∂H

∂ pi;

dpi

dt=−

∂H

∂xi . (5.1.9)


SincedH

dt= H ,H = 0,

it followsdHdt

= 0. Then: The fundamental function H(x, p) of a

Hamilton spaceHn is constant along the integral curves of the Hamil-ton - Jacobi equations (5.1.9). The Jacobi method of integration of(5.1.9) expound in chapter 4 is applicable and the variational prob-lem, in§4.2, ch. 4, can be formulated again in the case of Hamiltoniansystems(T∗M,θ ,H ).

5.2 Nonlinear connection of a Hamilton space

Let us consider a Hamilton spaceHn ⊂ (M,H(x, p)). The theory ofnonlinear connection on the manifoldT∗M, in ch. 4, can be applied inthe case of spacesHn. We must determine a nonlinear connectionNwhich depend only by the Hamilton spaceHn, i.e.N must be canonicalrelated toHn, as in the case of canonical nonlinear connection fromthe Lagrange spaces. To do this, direct method is given by using theLegendre transformation, suggested by Mechanics. We will present,in the end of this chapter, this method.

Now, following a result of R. Miron we enounce without demon-stration (which can be found in ch. 2,§2.3) the following result:Theorem 5.2.1.1The following set of functions

Ni j =14gi j ,H−

14

(gik

∂ 2H∂ pk∂x j +g jk

∂ 2H∂ pk∂xi

)(5.2.1)

are the coefficients of a nonlinear connection of the Hamilton spaceHn = (M,H(x, p)).

2The nonlinear connection N with coefficients Ni j , (5.2.1) dependsonly on the fundamental function H(x, p).

The brackets, from (5.2.1) are the Poisson brackets(,), §5.1.Indeed, by a straightforward computation, it follows that,under a

coordinate change( ), Ni j (x, p) from (5.2.1) obeys the rule of transfor-mation of the coefficients of a nonlinear connectionN. EvidentlyNi jdepend on the fundamental functionH(x, p) only. N− will be calledthe canonical nonlinear connection of the Hamilton spaceHn.

Remark 5.2.2.If the fundamental functionH(x, p) of Hn is globallydefined, onT∗M then the horizontal distribution determined by thecanonical nonlinear connectionN has the same property.

5.3 The canonical metrical connection of Hamilton spaceHn 85

It is not difficult to prove:

Proposition 5.2.1.The canonical nonlinear connection N has theproperties

τ i j =12(Ni j −Nji ) = 0 (it is symmetric) (5.2.2)

Ri jk +Rjki +Rki j = 0 (5.2.3)

Taking into account thatN is a horizontal distribution, we have theknown direct decomposition:

TuT∗M = Nu⊕Vu, ∀u∈ T∗M, (5.2.4)

it follows that (δ i , ∂ i) is an adapted basis to the previous splitting,where

δ i = ∂i −Ni j ∂ j (5.2.5)

and the dual basis(dxi ,δ pi) is:

δ pi = dpi −Ni j dxj . (5.2.5’)

Therefore we apply the theory to investigate the notion of metricN−linear connection determined only by the Hamilton spaceHn.

5.3 The canonical metrical connection of HamiltonspaceHn

Let us consider theN−linear connectionDΓ (N) = (H ijk,C

jki ) for

which N is the canonical nonlinear connection. By means of theoryfrom chapter 4 we can prove:

Theorem 5.3.1.In a Hamilton space Hn= (M,H(x, p)) there existsa unique N−linear connection DΓ (N) = (H i

jk,Cjki ) verifying the

axioms:N is the canonical nonlinear connection.

12The fundamental tensor gi j is h−covariant constant

gi j|k = δ kg

i j +gs jH isk+gisH j

sk= 0. (5.3.1)


3The fundamental tensor gi j is v−covariant constant

gi j |k = ∂ kCi j +gs jCiks +gisC jk

s = 0. (5.3.2)

4 DΓ (N) is h−torsion free:

T ijk = H i

jk −H ik j = 0 (5.3.3)

5DΓ (N) is v−torsion free:

Sjki =C jk

i −Ck ji = 0.

2) The N−connectiom DΓ (N) which verify the previous axioms hasthe coefficients Hijk and Cjk

i given by the coefficients Hijk and Cjki given

by the following generalized Christoffel symbols:

H ijk =

12

gis(δ jgsk+δ kg js−δ sg jk),

C jki =−

12

gis(∂ jgsk+ ∂ kg js−∂ sg jk).

(5.3.4)

Clearly CΓ (N) with the coefficients (5.3.4) is determined onlyby means of the Hamilton spaceHn. It is called canonical metricalN−linear connection.

Now, applying theorems from ch. 4 we have:Theorem 5.3.2.With respect to CΓ (N) we have the Ricci identities:

Xi| j |k−Xi

|k| j = XmR im jk−Xi|mRm jk,

Xi| j |

k−Xi|k| j = XmP i km j −Xi

|mC mkj −Xi |mP k

m j

Xi | j |k−Xi |k| j = XmS i jkm ,

(5.3.5)

where the d−tensors of torsion Ri jk ,Pijk and d tensors of curvature

R ij kh, P j k

i h , Sjkhi are expressed in chapter 4.

The structure equations, parallelism, autoparallel curves of theHamilton spaces are studied exactly as in chapter 4.

Example.The Hamilton space of electrodynamicsHn = (M,H(x, p))where the fundamental function is expressed in (5.1.4). Thefunda-

5.4 Generalized Hamilton SpacesGHn 87

mental tensor is1

mcgi j (x). The canonical nonlinear connection has

the coefficients

Ni j = γhi j (x)ph+

ec(Ai|k+Ak|i) (5.3.6)

whereγhi j (x) are the Christoffel symbols of the covariant tensor of

metric tensorgi j (x). EvidentlyAi|k = ∂kAi −Asγsik.

Remarking thatδgi j

δxk =∂gi j

∂xk we deduce that: the coefficients of

canonical metricalN−connectionHΓ (N) are:

H ijk(x) = γ i

jk(x), C jki = 0.

These geometrical object fieldsH,gi j , H ijk, C jk

i allow to develop thegeometry of the Hamilton spaces of electrodynamics.

5.4 Generalized Hamilton SpacesGHn

A straightforward generalization of the notion of Hamiltonspace isthat of generalized Hamilton space.

Definition 5.4.1.Ageneralized Hamilton space is a pairGHn = (M,gi j (x, p)) where M is a smooth realn−dimensional manifold andgi j (x, p) is a d−tensor field onT∗M of type (0,2) symmetric, non-degenerate and of constant signature.

The tensorgi j (x, p) is called fundamental. IfM a paracompact onT∗M there exist the tensorsgi j (x, p) which determine a generalizedHamilton space.

From the definition 5.1.1 it follows that: Any Hamilton spaceHn =(M,H) is a generalized Hamilton space.

The contrary affirmation is not true. Indeed, ifg

i j(x) is a Rieman-

nian tensor metric onM, then

gi j (x, p) = e−2σ (x,p)gi j(x), σ ∈ F (T∗M)

determine a generalized Hamilton space andgi j (x, p) does not the fun-damental tensor of an Hamilton space.


So, it is legitime the following definition:

Definition 5.4.2.A generalized Hamilton spaceGHn = (M,gi j (x, p))is called reducible to a Hamilton space if there exists a Hamilton func-tion H(x, p) on T∗M such that

gi j (x, p) =12

∂ i ∂ jH. (5.4.1)

Let us consider the Cartan tensor:

Ci jk =12

∂ ig jk. (5.4.2)

In a similar manner with the case of generalized Lagrange space (ch.2, §2.8) we have:

Proposition 5.4.2.A necessary condition that a generalized Hamiltonspace GHn = (M,gi j (x, p)) be reducible to a Hamilton one is that theCartan tensor Ci jk(x, p) be totally symmetric.

There exists a particular case when the previous condition is suffi-cient, too.

Theorem 5.4.1.A generalized Hamilton space GHn = (M,gi j (x, p))for which the fundamental tensor gi j (x, p) is 0-homogeneous is re-ducible to a Hamilton space, if and only if the Cartan tensor Ci jk(x, p)is totally symmetric.

Indeed, in this caseH(x, p) = gi j (x, p)pi p j is a solution of the equa-tion (5.4.1).

Let gi j (x, p) be the covariant of fundamental tensorgi j (x, p), thenthe following tensor field

C jki =−

12

gis(∂ jgsk+ ∂ kg js− ∂ sgsk) (5.4.3)

determine the coefficients of av−covariant derivative, which is met-rical, i.e.

gi j |k = ∂ kgi j +Ciks gs j+C jk

s gis = 0. (5.4.4)

The proof is very simple.We use thisv−derivation in the theory of canonical metrical con-

nection of the spacesGHn.In general, we cannot determine a nonlinear connection fromthe

fundamental tensorgi j of GHn. Therefore we study the geometry of

5.5 The almost Kahlerian model of a Hamilton space 89

spacesGHn when a nonlinear connectionN is a priori given. In thiscase we can apply the methods used in the construction of geometryof spacesHn.

Finally we remark that the class of spacesGHn include the class ofspacesHn:

GHn ⊃ Hn. (5.4.5)

5.5 The almost Kahlerian model of a Hamilton space

Let Hn = (M,H(x, p)) be a Hamilton space andgi j (x, p) its funda-mental tensor.

The canonical nonlinear connectionN has the coefficients (5.2.1).

The adapted basis to the distributionsN andV is

(δ

δxi= δ i = ∂i+

Ni j ∂ j , ∂ i)

and its dual basis(dxi ,δ pi = dpi −Nji dxj).

Thus, the following tensor on the cotangent manifoldT∗M =T∗M\0 can be considered:

G= gi j (x, p)dxi ⊗dxj +gi j δ pi ⊗δ p j . (5.5.1)

G determine a pseudo-Riemannian structure onT∗M. If the funda-mental tensorgi j (x, p) is positive defined, thenG is a Riemannianstructure onT∗M. G is called theN−lift of the fundamental tensorgi j . Clearly,G is determined by Hamilton spaceHN, only.

Some properties ofG:

1G is uniquely determined bygi j andNi j .2The distributionsN andV are orthogonal.

Taking into account the mappingF : X (T∗M)→X (T∗M) definedin (4.3.10) or, equivalently by:

F=−gi j ∂ i ⊗dxj +gi j δ i ⊗δ p j , (5.5.2)

one obtain:

Theorem 5.5.1.1 F is globally defined on the manifoldT∗M.2 F is an almost complex structure onT∗M:

F · F=−I . (5.5.3)


3 F depends on the Hamilton spaces Hn only.

Finally, one obtain a particular form of the Theorem 4.3.2:

Theorem 5.5.2.The following properties hold:

1The pair(G, F) is an almost Hermitian structure on the manifoldT∗M.

2The structure(G,F) is determined only by the fundamental functionH(x, p) of the Hamilton space Hn.

3The associated almost symplectic structure to(G, F) is the canoni-cal symplectic structureθ = dpi ∧dxi = δ pi ∧dxi .

4The space(T∗M,G, F) is almost Kahlerian.

The proof is similar with that for Lagrange space (cf. Ch. 3).The spaceH 2n = (T∗M,G, F) is called the almost Kahlerian

model of the Hamilton spaceHn. By means ofH 2n we can real-ize the study of gravitational and electromagnetic fields [176], [182],[183], [185], onT∗M.

Chapter 6Cartan spaces

A particular class of Hamilton space is given by the class of Cartanspaces. It is formed by the spacesHn = (M,H(x, p)) for which thefundamental functionH is 2−homogeneous with respect to momentapi . It is remarkable that these spaces appear as dual of the Finslerspaces, via Legendre transformations. Using this duality,several im-portant results in Cartan spaces can be obtained: the canonical non-linear connection, the canonical metrical connection etc.Therefore,the theory of Cartan spaces has the same symmetry and beauty likeFinsler geometry. Moreover, it gives a geometrical framework for theHamiltonian Mechanics or Physics fields.

The modern formulation of the notion of Cartan space is due ofR.Miron, but its geometry is based on the investigations of E. Cartan,A. Kawaguchi, H. Rund, R. Miron, D. Hrimiuc, and H. Shimada, P.L.Antonelli, S. Vacaru et. al. This concept is different from the notionof areal space defined by E. Cartan.

In the final part of this chapter we shortly present the notionofduality between Lagrange and Hamilton spaces.

6.1 Notion of Cartan space

Definition 6.1.1.A Cartan space is a pairCn = (M,K(x, p)) whereMis a realn−dimensional smooth manifold andK : T∗M → R is a scalarfunction which satisfies the following axioms:

1. K is differentiable onT∗M and continuous on the null section ofπ∗ : T∗M → M.

2. K is positive on the manifoldT∗M.3. K is positive 1−homogeneous with respect to the momentapi .4. The Hessian ofK2 having the components

91

92 6 Cartan spaces

gi j (x, p) =12

∂ i ∂ jK2 (6.1.1)

is positive defined on the manifoldT∗M.

It follows thatgi j (x, p) is a symmetric and nonsingulard−tensor fieldof type(0,2).

So, we haverank‖gi j (y, p)‖= n on T∗M. (6.1.2)

The functionsgi j (x, p) are 0−homogeneous with respect to mo-mentapi .

For a Cartan spaceC n = (M,K(x, p)) the functionK is called fun-damental function andgi j the fundamental or metric tensor.

If the baseM is paracompact, then on the manifoldT∗M there existsreal functionK such that the pair(M,K) is a Cartan space.

Indeed, onM there exists a Riemann structuregi j (x), x∈ M. Then

K(x, p) = gi j (x)pi p j1/2 (6.1.3)

determine a Cartan space.Other examples are given by

K = α∗+β ∗ (6.1.4)

K =(α∗)2

β(6.1.5)

whereα∗ = gi j (x)pi p j

1/2, β ∗ = bi(x)pi . (6.1.6)

K from (6.1.4) is called aRandersmetric andK from (6.1.5) is calleda Kropina metric.

A first and immediate result:

Theorem 6.1.1.If C n = (M,K) is a Cartan space then the pair HnC=

(M,K2) is an Hamilton space.

HnC

is called the associate Hamilton space withC n.This is the reason that the geometry of Cartan space include the

geometry of associate Hamilton space. So, we have the sequence ofinclusions

Rn ⊂ C n ⊂ Hn ⊂ GHn (6.1.7)

whereRn is the class of Riemann spacesRn = (M,gi j (x)) whichgive the Cartan spaces with the metric (6.1.3).

6.1 Notion of Cartan space 93

Now we can apply the theory from previous chapter.The canonical symplectic structureθ onT∗M:

θ = dpi ∧dxi (6.1.8)

andC n determine the Hamiltonian system(T∗M,θ ,K2). Then, setting

K =12

K2, (6.1.9)

we have:Theorem 6.1.2.For any Cartan spaceC n = (M,K(x, p)), the follow-ing properties hold:

1There exists a unique vector field XK2 on T∗M for which

iXK θ =−dK . (6.1.10)

2The vector field XK is expressed by

XK =∂K

∂ pi

∂∂xi −

∂K

∂xi

∂∂ pi

. (6.1.11)

3The integral curves of the vector field XK are given by the Hamilton–Jacobi equations:

dxi

dt=

∂K

∂ pi,

dpi

dt=−

∂K

∂xi . (6.1.12)

One deducedK

dt= K ,K = 0. So:

Proposition 6.1.1.The fundamental functionK =12

K2 of a Cartan

space is constant along the integral curves of the Hamilton–Jacobiequations (6.1.12).

The Jacobi method of integration of (6.1.12) can be applied.The Hamilton–Jacobi equations of a Cartan spaceC n are funda-

mental for the geometry ofC n. Therefore, the integral curves of thesystem of differential equations (6.1.12) are called the geodesics ofCartan spaceC n.

Other properties of the spaceC n:

1 pi =12

∂ iK2 is a 1-homogeneousd−vector field.

94 6 Cartan spaces

2gi j = ∂ j pi =12

∂ i ∂ jK2 is 0-homogeneous tensor field (the funda-

mental tensor ofC n).

3Ci jh =−14

∂ i ∂ j ∂ hK2 is (−1)−homogeneous with respect topi .

Proposition 6.1.2.We have the following identities:

pi = gi j p j , pi = gi j pj , (6.1.13)

K2 = gi j pi p j = pi pi , (6.1.14)

Ci jh ph =Cih j ph =Chi j ph = 0. (6.1.15)

Proposition 6.1.3.The Cartan spaceC n = (M,K(x, p)) is Rieman-nian if and only if the d−tensor Ci jh vanishes.

Considerd−tensor:

C jhi =−

12

gis∂ sg jh = gisCs jh. (6.1.16)

ThusC jhi are the coefficients of av−metric covariant derivation:

gi j |h = 0, Sjhi = 0

pi |k = δ k

i .(6.1.17)

6.2 Canonical nonlinear connection ofC n

The canonical nonlinear connection of a Cartan spaceC n = (M,K)is the canonical nonlinear connection of the associate Hamilton spaceHn

C= (M,K2). Its coefficientsNi j are given by (5.2.1).

Setting

γ ijh =

12

gis(∂ jgsh+∂hg js−∂sg jh) (6.2.1)

for the Christoffel symbols of the covariant of fundamentaltensor ofC n and using the notations

γ0jh = γ i

jhpi , γ0j0 = γ i

jhpi ph, (6.2.2)

we obtain:

6.3 Canonical metrical connection ofC n 95

Theorem 6.2.1 (Miron). The canonical nonlinear connection of theCartan spaceC n = (M,K(x, p)) has the coefficients

Ni j = γ0i j −

12

γ0h0∂ hgi j . (6.2.3)

The proof is based on the formula (5.2.1).Evidently, the canonical nonlinear connection is symmetric

Ni j = Nji . (6.2.4)

Let us consider the adapted basis(δ i , ∂ i) to the distributionsN (de-termined by canonical nonlinear connection) andV (vertical distribu-tion). We have

δ i = ∂i +Ni j ∂ j . (6.2.5)

The adapted dual basis(dxi ,δ pi) has the formsδ pi :

δ pi = dpi −Ni j dxj . (6.2.5’)

Thed−tensor of integrability of horizontal distributionN is

Ri jh = δ hNji −δ jNhi. (6.2.6)

By a direct calculus we obtain

Ri jh +Rjki +Rki j = 0. (6.2.7)

Proposition 6.2.4.The horizontal distribution N determined by thecanonical nonlinear connection of a Cartan spaceC n is integrableif and only if the d−tensor field Ri jk vanishes.

Proposition 6.2.5.The canonical nonlinear connection of a CartanspaceC n(M,K(x, p)) depends only on the fundamental function K(x,p).

6.3 Canonical metrical connection ofC n

Consider the canonical metricalN−linear connection of the HamiltonspaceHN

e = (M,K2). It is the canonical metricalN−linear connectionof the Cartan spaceC n = (M,K). It is denoted byCΓ (N) = (H i

jk,Cjki )

and shortly is named canonical metrical connection ofC n.

96 6 Cartan spaces

Then, theorem 5.3.1. implies:

Theorem 6.3.1.1) In a Cartan spaceC n = (M,K(x, p)) there existsa unique N−linear connection CΓ (N) = (H i

jk,Cjki ) verifying the ax-

ioms:

1N is the canonical nonlinear connection ofC n.2CΓ (N) is h−metrical

gi j|h = 0. (6.3.1)

3CΓ (N) is v−metrical:gi j |h = 0. (6.3.2)

4CΓ (N) is h−torsion free: Tijh = H i

jh−H ijh = 0.

5CΓ (N) is v−torsion free: Sjhi =C jh

i −Ch ji = 0.

2) The connection CΓ (N) has the coefficients given by the generalizedChristoffel symbols:

H ijh =

12

gis(δ jgsh+δ hg js−δ sg jh)

C jhi =−

12

gis(∂ jgsh+ ∂ hg js− ∂ sg jh).

(6.3.3)

3) CΓ (N) depends only on the fundamental function K of the CartanspaceC n.The connection CΓ (N) is called canonical metrical connection ofCartan spaceC n.

We denoteCΓ = (Ni j ,H ijk,C

jki ).

Evidently, thed−tensorC jhi has the properties (6.1.15) and the co-

efficientsCΓ have 1,0,−1 homogeneity degrees.

Proposition 6.3.6.The canonical metrical connections has the tensorof deflection:

∆i j = pi| j = 0, −δ ij = p j |

i = δ ij . (6.3.4)

But, the previous result allows to give a characterization of CΓ bya system of axioms of Matsumoto type:

Theorem 6.3.2.1 For any spaceC n = (M,K(x, p)) there exists aunique linear connection CΓ = (Ni j ,H i

jk,Cjki ) on the manifoldTM

verifying the following axioms:

6.3 Canonical metrical connection ofC n 97

1′. ∆i j = 0; 2′. gi j|h = 0; 3′. gi j |h = 0; 4′. T i

jh = 0; 5′. Sjhi = 0.

2 The previous metrical connection is exactly the canonical metri-cal connection CΓ .

The following properties ofCΓ (N) are immediately

1K|h = 0, K|h =ph

K;

2K2|h = 2ph;

3 pi| j = 0, pi |j = δ j

i ;4 pi

| j = 0, pi | j = gi j .

And thed−tensors of torsion ofCΓ (N) are

Ri jh = δ hNi j −δ jNih,Cjhi ,T i

jh = 0,Sjhi = 0,Pi

jh = H ijh− ∂ iNjh.

(6.3.5)Of course, we have

Ri jh =−Rih j ,Pijh = Pi

h j, C jhi =Ch j

i . (6.3.6)

Proposition 6.3.7.The d−tensor of curvature Si jhk is given by:

Si jhk =Cmh

k Ci jm−Cm j

k Cihm. (6.3.7)

Denoting byRi j

kh = gisR js kh, etc.

and applying the Ricci identities (5.3.5), one obtains:

Theorem 6.3.3.The canonical metrical connectionCΓ (N) of the Car-tan spaceC n satisfies the identities:

Ri jkh+Rji

kh = 0, Pi j hk +P ji h

k = 0, Si jkh+Sjikh = 0. (6.3.8)

R oi jk +Ri jk = 0, P o k

i j +Pki j = 0, So jk

i = 0. (6.3.9)

Ro jk = 0, Pko j = 0. (6.3.10)

Of course, the index “o” means the contraction bypi or pi .The 1-form connections ofCΓ (N) are:

ω ij = H i

jhdxh+Cihj δ ph. (6.3.11)

98 6 Cartan spaces

Taking into account (6.2.5’), one obtains

Theorem 6.3.4.The structure equations of the canonical metricalconnection CΓ (N) of the Cartan spaceC n = (M,K(x, p)) are

d(dxi)−dxm∧ω im =−Ω i ;

d(δ pi)+δ pm∧ωmi =−

Ω i ;

dω ij −ωm

j ∧ω im =−Ω i

j ,

(6.3.12)

Ω i ,

Ω i being the 2-forms of torsion:

Ω i =Cikj dxj ∧δ pk

Ω i =

12

Ri jkdxj ∧dxk+Pki j dxj ∧δ pk

(6.3.13)

andΩ ij is 2-form of curvature:

Ω ij=

12

Rij kmdxk∧dxm+P i m

j k dxk∧δ pm+12

S ikmj δ pk∧δ pm. (6.3.14)

Applying Proposition 4.4.2, we determine the Bianchi identities ofCΓ (N).

Now, we can develop the geometry of the associated HamiltonspaceHn

C= (M,K2(x, p)). Also, in the case of Cartan space, the geo-

metrical modelK 2n = (T∗M,G,F) is an almost Kahlerian one.Before finish this chapter is opportune to say some words on the

Legendre transformation.

6.4 The duality between Lagrange and Hamiltonspaces

The duality, via Legendre transformation, between Lagrange and Hamil-ton space was formulated by R. Miron in the papers and it was devel-oped by D. Hrimiuc, P.L. Antonelli, D. Bao, et al. Of course, it wassuggested by Theoretical Mechanics.

The theory of Legendre duality is presented here follows theChap-ter 7 of the book [174].

6.4 The duality between Lagrange and Hamilton spaces 99

Let L be a regular Lagrangian,L =12

L on a domainD ⊂ TM and

let H be a regular HamiltonianH =12

H, on a domainD∗ ⊂ T∗M.

Hence for∂i =∂

∂yi , ∂ i =∂

∂ pithe matrices with entries:

gi j (x,y) = ∂i ∂ jL (x,y),

g∗i j (x, p) = ∂ i ∂ jH (x, p)(6.4.1)

are nondegenerate onD and onD∗, (i, j, ...= 1,2, ...,n).SinceL ∈ F (D) is a differentiable map consider thefiber deriva-

tive of L locally given by

ϕ(x,y) = (xi , ∂ jL (x,y)) (6.4.2)

which is called theLegendre transformation. It is easy to see that:L isa regular Lagrangian if and only ifϕ is a local diffeomorphism, [174].In the same manner, forH ∈ FD∗, the fiber derivative is locallygiven by

ψ(x,y) = (xi , ∂ iH (x,y)), (6.4.3)

which is a local diffeomorphism if and only ifH is regular.

Now, let us consider12

L(x,y) = L (x,y) the fundamental function

of a Lagrange space. Thenϕ defined by (6.4.2) is a diffeomorphismbetween two open setU ⊂D andU∗⊂D∗. In this case, we can define

H∗(x, p) = piy

i −L (x,y), (6.4.4)

wherey= (yi) is solution of the equation

pi = ∂iL (x,y). (6.4.4’)

Remark 6.4.1.In the theory of Lagrange space the functionE (x, p) =yi ∂iL −L (x,y) is the energy of Lagrange spaceLn.

It follows without difficulties thatH∗n = (M,H∗(x, p)), H∗ = 2H ∗

is a Hamilton space. Its fundamental tensorg∗i j (x, p) is given bygi j (x,ϕ−1(x, p)).

We setH∗n = Leg Ln and say thatH∗n is the dual ofLn via Legendretransformation determined byLn.

100 6 Cartan spaces

Remark 6.4.2.The mappingLeg was used in chapter 2 to transformthe Euler–Lagrange equations (ch. 2, part I) into the Hamilton–Jacobiequations (ch. 2, part I).

Analogously, for a Hamilton spaceHn = (M,H(x, p)), H =12

H,

consider the function

L∗(x,y) = piy

j −H (x, p),

where p = (pi) is the solution of the equation (6.4.4). ThusL∗n =(M,L∗(x,y)), L∗ = 2L ∗, is a Lagrange space, dual, via Legendretransformation of the Hamilton spaceHn. So,L∗n = Leg Hn.

One proves:Leg(Leg Ln) = Ln, Leg(Leg Hn) = Hn.The diffeomorphismsϕ and ψ have the propertyϕ = ψ−1. And

they transform the fundamental object fields fromLn in the funda-mental object fields ofH∗n = Leg Ln, and conversely. For instance,the Euler–Lagrange equation ofLn are transformed in the Hamilton–Jacobi equations ofHn. The canonical nonlinear connection ofLn istransformed in the canonical nonlinear connection ofH∗n etc.

Examples.1 The Lagrange space of ElectrodynamicsLn

0 = (M,L0(x,y)):

L0(x,y) = mcγ i j (x)yiy j +

2em

Ai(x)yi

whereγ i j (x) is a pseudo-Riemannian metric, has the Legendre trans-formation:

ϕ : xi = xi , pi =12

∂L0

∂yi = mcγ i j (x)y+em

Ai(x)

and

ϕ−1 = ψ : xi = xi , yi =1

mcγ i j (x)

(p j −

em

A j(x)).

The Hamilton spaceH∗0 = Leg L0 has the fundamental function:

H∗0 =

1mc

γ i j (x)pi p j −2e

mc2Ai(x)pi +e2

mc3Ai(x)A j(x). (6.4.5)

2 The Lagrange space of ElectrodynamicsLn = (M,L(x,y)) inwhichL = L0+U(x), ((2.1.4, Ch. 2)), i.e.

6.4 The duality between Lagrange and Hamilton spaces 101


2em

Ai(x)yi +U(x),

has the Legendre transformation:

ϕ i : xi = xi , pi = mcγ i j (x)yj +

em

Ai(x)

ϕ−1 = ψ : xi = xi , yi =1

mcγ i j (x)

(p j −

em

A j

).

And H∗n = Leg Ln has the fundamental functionH∗(x, p) given by

H∗(x, p) =1

c(m+ Uc2 )

γ i j (x)pi p j −2e

c2(m+ Uc2)

piAi(x)+

+e2

c3(m+ Uc2)

Ai(x)Ai(x)−

1c

U(x)c

.

3 The class of Finsler spaceFn is inclosed in the calss of La-grange spaceLn, that isFn ⊂ LN, one can consider the re-striction of the Legendre transformationLeg Ln to the class of Finslerspaces.

In this case, the mappingϕ : (x,y) ∈ D → (x, p) ∈ D∗ is given by

ϕ(x,y) = (x, p) with

pi =12

∂ iF2 (6.4.6)

and one obtains:Leg(Fn) = C ∗n, C ∗n = (M,K∗(x, p)) with

K∗2 = 2piyi −F2(x,y) = yi ∂iF

2(x,y) = F2(x,y)

andyi is solution of the equation (6.4.6).

Theorem 6.4.1.The dual, via Legendre transformation, of a Finslerspace Fn=(M,F(x,y)) is a Cartan spaceC ∗n=(M,F(x,ϕ−1(x, p))).

Part IILagrangian and Hamiltonian Spaces of

higher order

In this part of the book we study, the notions of Lagrange and Hamil-ton spaces of orderk. They was introduced and investigated by theauthor [161]. Without explicitly formulated a clear definition of thesespaces, a major contributions to edifice of these geometricshave beendone by M. Crampin and colab. [64], M. de Leone and colab. [138],A. Kawaguchi [120], I. Vaisman [254] etc.

For details, we refer to the books:The Geometry of Higher Or-der Lagrange Spaces. Applications to Mechanics and Physics, KluwerAcad. Publ. FTPH, 82, 1997, [161];The Geometry of Higher–OrderFinsler Spaces, Hadronic Press, Inc. USA, 1998, [162];The Geometryof Higher Order Hamilton Spaces. Applications to Hamiltonian Me-chanics, Kluwer Acad. Publ., FTPH 132, 2003 [163]; as well as thepapers [167]–[172].

This part contents: The geometry of the manifold of accelerationsTkM, Lagrange spaces of orderk, L(k)n, Finsler spaces of orderk,F(k)n and dual, via Legendre transformation Hamilton spacesH(k)n

and Cartan spacesC (k)n.

Chapter 1The Geometry of the manifoldTkM

The importance of Lagrange geometries consists of the fact that thevariational problems for Lagrangians have numerous applications:Mathematics, Physics, Theory of Dynamical Systems, Optimal Con-trol, Biology, Economy etc.

But, all of the above mentioned applications have imposed also theintroduction of the notions of higher order Lagrange spaces. The basemanifold of this space is the bundle of accelerations of superior or-der. The methods used in the construction of the geometry of higherorder Lagrange spaces are the natural extensions of those used in theedification of the Lagrangian geometries exposed in chapters 1, 2 and3.

The concept of higher order Lagrange space was given by authorin the books [161], [155]. The problems raised by the geometriza-tion of Lagrangians systems of orderk > 1 were been investigatedby many scholars: Ch. Ehresmann [82], P. Libermann [143], J.Pom-maret [?], J. T. Synge [243], M. Crampin [64], P. Saunders [230], G.S.Asanov [27], D. Krupka [132], M. de Leon [141], H. Rund [218], A.Kawaguchi [119], K. Yano [256], K. Kondo [128], D. Grigore [92],R. Miron [155], [156] et al.

In this chapter we shall present, briefly, the following problems:

1The geometry of total space of the bundle of higher order accelera-tions.

2The definition of higher order Lagrange space, based on the nonde-generate Lagrangians of orderk≥ 1.

3The solving of the old problem of prolongation of the Riemannianstructures, given on the base manifoldM, to the Riemannian struc-tures on the total space of the bundle of accelerations of orderk≥ 1,we prove for the first time the existence of Lagrange spaces oforderk≥ 1.

4The elaboration of the geometrical ground for variational calculusinvolving Lagrangians which depend on higher order accelerations.

5The introduction of the notion of higher order energies and proof ofthe law of conservation.

105

106 1 The Geometry of the manifoldTkM

6The notion ofk−semispray. Nonlinear connection the canonicalmetrical connection and the structure equations.

7The Riemannian(k− 1)n−almost contact model of a Lagrangespace of orderk.

Evidently, we can not sufficiently develop these subjects. For muchmore informations one can see the books [161], [162].

Throughout in this chapter the differentiability of manifolds and ofmappings means the classC∞.

1.1 The bundle of acceleration of orderk≥ 1

In Analytical Mechanics a realn−dimensional differentiable mani-fold M is considered as space of configurations of a physical system.A point (xi) ∈ M is called a material point. A mappingc : t ∈ I →(xi(t)) ∈ U ⊂ M is a law of moving (a law of evolution),t is time, a

pair (t,x) is an event and thek-uple

(dxi

dt, · · · ,

1k!

dkxi

dtk

)gives the ve-

locity and generalized accelerations of order 2, ...,k−1. The factors1h!

(h= 1, ...,k) are introduced here for the simplicity of calculus. In

this chapter we omit the word “generalized” and say shortly,the ac-

celeration of orderh, for1h!

dhxi

dth. A law of movingc : t ∈ I → c(t)∈U

will be called a curve parametrized by timet.In order to obtain the differentiable bundle of accelerations of or-

der k, we use the accelerations of orderk, by means of geometricalconcept of contact of orderk between two curves in the manifoldM.

Two curvesρ ,σ : I → M in M have at the pointx0 ∈ M, ρ(0) =σ(0) = x0 ∈U (andU a domain of local chart inM) have acontactoforderk if we have

dα( f ρ)(t)dtα

|t=0 =dα( f σ)(t)

dtα|t=0,(α = 1, ...,k). (1.1.1)

It follows that: the curvesρ andσ have at the pointx0 = ρ(0) =σ(0) a contact of orderk if and only if the accelerations of order1,2, ...,k on the curveρ at x0 have the same values with the corre-sponding accelerations on the curveσ at pointx0.

The relation “to have a contact of order k” is an equivalence. Let[ρ ]x0 be a class of equivalence andTk

x0M the set of equivalence classes.

Consider the set

1.1 The bundle of acceleration of orderk≥ 1 107

TkM =⋃

x0∈M

Tkx0

M (1.1.2)

and the mapping:

πk : [ρ]x0 ∈ TkM → x0 ∈ M, ∀[ρ ]x0. (1.1.2’)

Thus the triple(TkM,πk,M) can be endowed with a natural differ-entiable structure exactly as in the casesk= 1, when(T1M,π1,M) isthe tangent bundle.

If U ⊂ M is a coordinate neighborhood on the manifoldM, x0 ∈Uand the curveρ : I →U , ρ0 = x0 is analytical represented onU by theequationsxi = xi(t), t ∈ I , thenTk

x0M can be represented by:

xi0 = xi(0), y(1)i0 =

dxi

dt(0), ..., y(k)i0 =

1k!

dkxi

dtk(0). (1.1.3)

Setting

φ : ([ρ ]x0) ∈ TkM → φ([ρ ]x0) = (xi0,y

(1)i0 , ...,y(k)i0 ) ∈ R(k+1)n, (1.1.4)

it follows that the pair(πk)−1(U),φ) is a local chart onTkM inducedby the local chart(U,ϕ) on the manifoldM.

So a differentiable atlas of the manifoldM determine a differen-tiable atlas onTkM and the triple(TkM,πk,M) is a differentiablebundle. Of course the mappingπk : TkM → M is a submersion.

(TkM,πk,M) is called thek accelerations bundle or tangent bun-dle of orderk or k−osculator bundle. A change of local coordinates(xi ,y(1)i , ...,y(k)i) → (xi , y(1)i , ..., y(k)i) on the manifoldTkM, accord-ing with (1.1.3), is given by:


xi = xi(x1, ...,xn), rank

(∂ xi

∂x j

)= n

y(1)i =∂ xi

∂x j y(1) j

2y(2)i =∂ y(1)i

∂x j y(1) j +2∂ y(1)i

∂y(1) jy(2) j

.................................................

ky(k)i=∂ y(k−1)i

∂x j y(1) j+2∂ y(k−1)i

∂y(1) jy(2) j+...+k

∂ y(k−1)i

∂y(k−1) jy(k) j .

(1.1.5)

And remark that we have the following identities:

∂ y(α)i

∂x j =∂ y(α+1)i

∂y(1) j= ...=

∂ y(k)i

∂y(k−α) j,

(α = 0, ...,k−1;y(0) = x).

(1.1.5’)

We denote a pointu∈ TkM by u= (x,y(1), ...,y(k)) and its coordi-nates by(xi ,y(1)i , ...,y(k)i).

A section of the projectionπk is a mappingS: M → TkM with thepropertyπkS= 1M. And a local sectionShas the propertyπkS|U =1U .

If c : I → M is a smooth curve, locally represented byxi = xi(t),t ∈ I , then the mappingc : I → TkM given by:

xi = xi(t), y(1)i =11!

dxi

dt(t), ..., y(k)i =

1k!

d(k)xi

dtk(t), t ∈ I (1.1.6)

is the extension of order kto TkM of c. We haveπk c= c.The following property holds:If the differentiable manifoldM is paracompact, thenTkM is a para-

compact manifold.We shall use the manifoldTM = TkM \ 0, where 0 is the null

section ofπk.

1.2 The Liouville vector fields 109

1.2 The Liouville vector fields

The natural basis at pointu∈ TkM of Tu(TkM) is given by(

∂∂xi ,

∂∂y(1)i

, ...,∂

∂y(k)i

)

u.

A local coordinate changing (1.1.5) transform the natural basis by thefollowing rule:

∂∂xi =

∂ x j

∂xi

∂∂ x j +

∂ y(1) j

∂xi

∂∂ y(1) j

+ ...+∂ y(k) j

∂xi

∂∂ y(k) j

∂∂y(1)i

=∂ y(1) j

∂y(1)i∂

∂ y(1) j+ ...+

∂ y(k) j

∂y(1)i∂

∂ y(k) j

...............................................................∂

∂y(k)i=

∂ y(k) j

∂y(k)i∂

∂ y(k) j,

(1.2.1)

calculated at the pointu∈ TkM.The natural cobasis(dxi ,dy(1)i , ...,dy(k)i)u is transformed by (1.1.5)

as follows:

dxi =∂ xi

∂x j dxj ,

dy(1)i =∂ y(1)i

∂x j dxj +∂ y(1)i

∂y(1) jdy(1) j ,

...............................................................

dy(k)i =∂ y(k)i

∂x j dxj +∂ y(k)i

∂y(1) jdy(1) j + ...+

∂ y(k)i

∂y(k) jdy(k) j .

(1.2.1’)

The formulae (1.2.1) and (1.2.1’) allow to determine some impor-tant geometric object fields on the total space of accelerations bundleTkM.

The vertical distributionV1 is local generated by the vector fields∂

∂y(1)i, ...,

∂∂y(k)i

, i = 1, ...,n. V1 is integrable and of dimension

kn. The distributionV2 local generated by

∂

∂y(2)i, ...,

∂∂y(k)i

is also


integrable, of dimension(k−1)n and it is a subdistribution ofV1. Andso on.

The distributionVk local generated by

∂

∂y(k)i

is integrable and

of dimensionn. It is a subdistribution of the distributionVk−1. So wehave the following sequence:

V1 ⊃V2 ⊃ ...⊃Vk

Using again (1.2.1) we deduce:

Theorem 1.2.1.The following operators in the algebra of functionsF (TkM) :

1Γ= y(1)i

∂∂y(k)i

,

2Γ= y(1)i

∂∂y(k−1)i

+2y(2)i∂

∂y(k)i,

..............................................k

Γ= y(1)i∂

∂y(1)i+2y(2)i

∂∂y(2)i

+ ...+ky(k)i∂

∂y(k)i

(1.2.2)

are the vector fields on TkM. They are independent on the manifold

TkM and1

Γ⊂Vk,2

Γ⊂Vk−1,...,k

Γ⊂V1.

The vector fields1

Γ ,2

Γ , ...,k

Γ are called theLiouville vector fields.Also we have:

Theorem 1.2.2.For any function L∈ F (TkM), the following entries

are1-form fields on the manifoldTkM:

d0L =∂L

∂y(k)idxi ,

d1L =∂L

∂y(k−1)idxi +

∂L

∂y(k)idy(1)i,

..............................................

dkL =∂L∂xi dxi +

∂L

∂y(1)idy(1)i + ...+

∂L

∂y(k)idy(k)i.

(1.2.3)

Evidently,dkL = dL.

1.2 The Liouville vector fields 111

In applications we shall use also the following nonlinear operator

Γ = y(1)i∂

∂xi +2y(2)i∂

∂y(1)i+ ...+ky(k)i

∂∂y(k−1)i

. (1.2.4)

Γ is not a vector field onTkM.

Definition 1.2.1.A k-tangent structureJ on TkM is the F (TkM)-linear mappingJ : X (TkM)→ X (TkM), defined by:

J

(∂

∂xi

)=

∂∂y(1)i

, J

(∂

∂y(1)i

)=

∂∂y(2)i

, ...,

J

(∂

∂y(k−1)i

)=

∂∂y(k)i

, J

(∂

∂y(k)i

)= 0.

(1.2.5)

It is not difficult to see thatJ has the properties:

Proposition 1.2.1.We have:1. J is globally defined on TkM,2. J is an integrable structure,3. J is locally expressed by

J =∂

∂y(1)i⊗dxi +

∂∂y(2)i

⊗dy(1)i + ...+∂

∂y(k)i⊗dy(k−1)i (1.2.6)

4. ImJ= KerJ, KerJ=Vk,5. rankJ= kn,

6. Jk

Γ=k−1Γ , ..., J

2Γ=

1Γ , J

1Γ= 0,

7. JJ ...J= 0, (k+1 factors).

In the next section we shall use the functions

I1(L) = L 1Γ

L, ..., I k(L) = L kΓ

L, ∀L ∈ F (TkM), (1.2.7)

whereL αΓ

is the operator of Lie derivation with respect to the Liou-

ville vector fieldα

Γ .The functionsI1(L), ..., I k(L) are called themain invariantsof the

functionL. They play an important role in the variational calculus.


1.3 Variational Problem

Definition 1.3.1.A differentiable Lagrangian of orderk is a mapping

L : (x,y(1), ...,y(k))∈TkM → L(x,y(1),y(k))∈R, differentiable onTkM

and continuous on the null section 0 :M → TkM of the projectionπk : TkM → M.

If c : t ∈ [0,1]→ (xi(t))∈U ⊂M is a curve, with extremitiesc(0)=

(xi(0)) and c(1) = (xi(1)) and c : [0,1] → TkM from (1.1.6) is itsextension. Then the integral of action ofL c is defined by

I(c) =∫ 1

0L

(x(t),

dxdt

(t), ...,1k!

dkx

dtk(t)

)dt. (1.3.1)

Remark 1.3.1.One proves that ifI(c) does not depend on the parametriza-tion of curvec then the following Zermelo conditions holds:

I1(L) = ...= I k−1(L) = 0, I k(L) = L. (1.3.2)

Generally, these conditions are not verified.The variational problem involving the functionalI(c) from (1.3.1)

will be studied as a natural extension of the theory expounded in §2.2,Ch. 2.

On the open setU we consider the curves

cε : t ∈ [0,1]→ (xi(t)+ εV i(t)) ∈ M, (1.3.3)

whereε is a real number, sufficiently small in absolute value suchthat Imcε ⊂ U , V i(t) = V i(x(t)) being a regular vector field onU ,restricted toc. We assume all curvescε have the same end pointsc(0)andc(1) and their osculator spaces of order 1,2, ...,k−1 coincident atthe pointsc(0), c(1). This means:

V i(0) =V i(1) = 0;dαV i

dtα(0) =

dαV i

dtα(1) = 0,

(α = 1, ...,k−1).

(1.3.3’)

The integral of actionI(cε) of the LagrangianL is:

I(cε)=∫ 1

0L

(x+ εV,

dxdt

+ εdVdt

, ...,1k!

(dkx

dtk+ ε

dkV

dtk

))dt. (1.3.4)

1.3 Variational Problem 113

A necessary condition forI(c) to be an extremal value forI(cε) is

dI(cε)

dε|ε=0 = 0. (1.3.5)

Thus, we have

dI(cε)

dε=

∫ 1

0

ddε

L

(x+ εV,

dxdt

+ εdVdt

, ...,1k!

(dkx

dtk+ ε

dkV

dtk

))dt.

The Taylor expansion ofL for ε = 0, gives:

dI(cε)

dε|ε=0=

∫ 1

0

(∂L∂xi V

i+∂L

∂y(1)idVi

dt+...+

1k!

∂L

∂y(k)idkV i

dtk

)dt. (1.3.6)

Now, with notations

Ei (L) :=

∂L∂xi −

ddt

∂L

∂y(1)i+ ...+(−1)k 1

k!dk

dtk∂L

∂y(k)i(1.3.7)

and

I1VL =V i ∂L

∂y(k)i, I2

V(L) =V i ∂L

∂y(k−1)i+

dVi

dt∂L

∂y(k)i, ...,

I kV =V i ∂L

∂y(1)i+

dVi

dt∂L

∂y(2)i+ ...+

1(k−1)!

dk−1Vv

dtk−1

∂L

∂y(k)i

(1.3.8)

we obtain an important identity:

∂L∂xi V

i +∂L

∂y(1)idVi

dt+ ...+

1k!

∂L

∂y(k)idkV i

dtk=

Ei (L)+

+ddt

I kV(L)−

12!

ddt

I k−1V (L)+...+(−1)k−1 1

k!dk−1

dtk−1 I1V(L)

(1.3.9)

Now, applying (1.3.7) and taking into account (1.3.3’) with

IαV (L)(c(0)) = Iα

V (L)c(1) = 0, (α = 1,2, ...,k)

we obtain


dI(cε)

dε|ε=0 =

∫ 1

0

0Ei (L)V

idt. (1.3.10)

But V i(t) is an arbitrary vector field. Therefore the equalities (1.3.5)and (1.3.10) lead to the following result:

Theorem 1.3.1.In order that the integral of action I(c) be an extremalvalue for the functionals I(cε), (1.3.4) is necessary that the followingEuler - Lagrange equations hold:

0Ei (L) :=

∂L∂xi −

ddt

∂L

∂y(1)i+...+(−1)k 1

k!dk

dtk∂L

∂y(k)= 0,

y(1)i =dxi

dt, · · · ,y(k)i =

1k!

dkxi

dtk.

(1.3.11)

One proves [161] that0Ei (L) is a covector field. Consequence the

equation0Ei (L) = 0 has a geometrical meaning.

Consider the scalar field

Ek(L)=I k(L)−

12!

ddt

I k−1(L)+...+(−1)k−1 1k!

dk−1

dtk−1 I1(L)−L.

(1.3.12)It is called theenergy of order kof the LagrangianL.

The next result is known, [161]:

Theorem 1.3.2.For any Lagrangian L(x,y(1), · · · ,y(k)) the energy oforder k,E k(L) is conserved along every solution curve of the Euler -

Lagrange equations0Ei (L) = 0, y(1)i =

dxi

dt, · · · ,y(k)i =

1k!

dkxi

dtk.

Remark 1.3.2.Introducing the notion of energy of order 1,2, · · · ,k−1we can prove a Nother theorem for the Lagrangians of orderk.

Now we remark that for anyC∞−function φ(t) and any differen-tiable LagrangianL(x,y(1), · · · ,y(k)) the following equality holds:

0Ei (φL) = φ

0Ei (L)+

dφdt

1Ei (L)+ · · ·+

dkφdtk

kEi (L), (1.3.13)

where1Ei (L), · · · ,

kEi (L) ared-covector fields - called Graig - Synge

covectors, [63]. We consider the covectork−1Ei (L):

1.3 Variational Problem 115

k−1E i (L) = (−1)k−1 1

(k−1)!

(∂L

∂y(k−1)i−

ddt

∂L

∂y(k)i

), (1.3.14)

It is important in the theory ofk−semisprays from the Lagrangespaces of orderk.

The Hamilton - Jacobi equations, of a spaceLn = (M,L(x,y)) in-troduced in the section 4 of Chapter 2 can be extended in the higherorder Lagrange spaces by using the Jacobi - Ostrogradski momenta.Indeed, the energy of orderk, E k(L) from (1.3.13) is a polynomial

function indxi

dt, · · · ,

dkxi

dtkgiven by

Ek(L) = p(1)i

dxi

dt+ p(2)i

d2xi

dt2+ · · ·+ p(k)i

dkxi

dtk−L, (1.3.15)

where

p(1)i =∂L

∂y(1)i−

12!

ddt

∂L

∂y(2)i+...+(−1)k−1 1

k!dk−1

dtk−1

∂L

∂y(k)i,

p(2)i=12!

∂L

∂y(2)i−

13!

ddt

∂L

∂y(3)i+...+(−1)k−2 1

k!dk−2

dtk−2

∂L

∂y(k)i,

.......................................................................

p(k)i =1k!

∂L

∂y(k)i.

(1.3.16)

p(1)i , ..., p(k)i are calledthe Jacobi-Ostrogradski momenta.

The following important result has been established by M. deLeonand others, [161]:

Theorem 1.3.3.Along the integral curves of Euler Lagrange equa-

tions0Ei (L) = 0 the following Hamilton - Jacobi - Ostrogradski equa-

tions holds:


∂E k(L)∂ p(α)i

=dαxi

dtα, (α = 1, ...,k),

∂E k(L)∂xi =−

dp(1)idt

,

1α!

∂E k(L)

∂y(α)i=−

dp(α+1)i

dt, (α = 1, ...,k−1).

Remark 1.3.3.The Jacobi - Ostrogradski momenta allow the introduc-tion of the 1-forms:

p(1) = p(1)idxi + p(2)idy(1)i + · · ·+ p(k)idy(k−1)i ,

p(2) = p(2)idxi + p(3)idy(1)i + · · ·+ p(k)idy(k−2)i ,..............................................................p(k) = p(k)idxi .

1.4 Semisprays. Nonlinear connections

A vector fieldS∈ χ(TkM) with the property

JS=k

Γ (1.4.1)

is called ak−semispray onTkM. S can be uniquely written in theform:

S= y(1)i∂

∂xi + ...+ky(k)i∂

∂y(k−1)i− (k+1)Gi(x,y(1), ...,y(k))

∂∂y(k)i

,

(1.4.2)or shortly

S= Γ − (k+1)Gi ∂∂y(k)i

. (1.4.2’)

The set of functionsGi is the set ofcoefficients of S. With respectto (1.1.5) Ch. 4Gi are transformed as following:

(k+1)Gi = (k+1)G j ∂ xi

∂x j −Γ y(k)i . (1.4.3)

1.4 Semisprays. Nonlinear connections 117

A curve c : I → M is called ak-pathon M with respect toS if itsextensionc is an integral curve ofS. A k-path is characterized by the(k+1)− differential equations:

dk+1xi

dtk+1 +(k+1)Gi(

x,dxdt

, ...,1k!

dkx

dtk

)= 0. (1.4.4)

We shall show that ak−semispray determine the main geometricalobject fields onTkM as: the nonlinear connectionsN, theN−linearconnectionsD and them structure equations. Evidently,N andD arebasic for the geometry of manifoldTkM.

Definition 1.4.1.A subbundleHTkM of the tangent bundle(TTkM,dπk, TkM) which is supplementary to the vertical subbundleV1TkM:

TTkM = HTkM⊕V1TkM (1.4.5)

is called anonlinear connection.

The fibres ofHTkM determine a horizontal distribution

N : u∈ TkM → Nu = HuTkM ⊂ TuTkM, ∀u∈ TkM

supplementary to the vertical distributionV1, i.e.

TuTkM = Nu⊕V1,u, ∀u∈ TkM. (1.4.5’)

If the base manifoldM is paracompact onTkM there exist the non-linear connections.

The local dimension ofN is n= dimM.Consider a nonlinear connectionN and denote byh andv the hori-

zontal and vertical projectors with respect toN andV1:

h+v= I , hv= vh= 0, h2 = h, v2 = v.

As usual we denote

XH = hX, XV = vX, ∀X ∈ χ(TkM).

An horizontal lift, with respect toN is aF (M)-linear mappinglh :X (M)→ X (TkM) which has the properties

v lh = 0, dπk lh = Id.


There exists an unique local basis adapted to the horizontaldistri-butionN. It is given by

δδxi = lh

(∂

∂xi

),(i = 1, ...,n). (1.4.6)

The linearly independent vector fields of this basis can be uniquelywritten in the form:

δδxi =

∂∂xi −N j

i(1)

∂∂y(1) j

− ...−N ji

(k)

∂∂y(k) j

. (1.4.7)

The systems of differential functions onTkM : (N ji

(1), ...,N j

i(k)

), gives

thecoefficientsof the nonlinear connectionN.By means of (1.4.6) it follows:

Proposition 1.4.1.With respect of a change of local coordinates onthe manifold TkM we have

δδxi =

∂ x j

∂xi

δδ x j , (1.4.7’)

and

Nim

(1)

∂ xm

∂x j = Nmj

(1)

∂ xi

∂xm −∂ y(1)i

∂x j ,

..............................................................

Nim

(k)

∂ xm

∂x j = Nmj

(k)

∂ xi

∂xm + ...+Nmj

(1)

∂ y(k−1)i

∂xm −∂ y(k)i

∂x j ,

(1.4.8)

Remark 1.4.1.The equalities (1.4.8) characterize a nonlinear connec-tion N with the coefficientsN j

i(1)

, · · · ,N ji

(k)

.

These considerations lead to an important result, given by:

Theorem 1.4.1 (I. Bucataru [47]). If S if a k−semispray on TkM,with the coefficients Gi, then the following system of functions:

1.4 Semisprays. Nonlinear connections 119

Nj(1)

i =∂Gi

∂y(k)i,Nj(2)

i =∂Gi

∂y(k−1)i, · · · ,Nj

(k)

i =∂Gi

∂y(1) j(4.9)

gives the coefficients of a nonlinear connection N.

The k−tangent structureJ, defined in (1.2.5), Ch. 4. applies thehorizontal distributionN0 = N into a vertical distributionN1 ⊂ V1 ofdimensionn, supplementary to the distributionV2. Then it applies thedistributionN1 in distributionN2 ⊂ V2, supplementary to the distri-bution V3 and so on. Of course we have dimN0 = dimN1 = · · · =dimNk−1 = n.

Therefore we can write:

N1 = J(N0), N2 = J(N1), · · · ,Nk−1 = J(Nk−2) (1.4.9)

and we obtain the direct decomposition:

TuTkM = N0,u⊕N1,u⊕·· ·⊕Nk−1,u⊕Vk,u, ∀u∈ TkM. (1.4.10)

An adapted basis toN0, N1, ...,Nk−1, Vk is given by:

δδxi ,

δδy(1)i

, · · · ,δ

δy(k−1)i,

∂∂y(k)i

,(i = 1, ...,n), (1.4.11)

whereδ

δxi is in (1.4.7) and

δδy(1)i

=∂

∂y(1)i−N j

i(1)

∂∂y(2)i

−·· ·− N ji

(k−1)

∂∂y(k)i

,

..................................................................δ

δy(k−1)i=

∂∂y(k−1)i

−N ji

(1)

∂∂y(k) j

.

(1.4.12)

With respect to (1.1.5), Ch. 4 we have:

δδy(α)i

=∂x j

∂ xi

δδ y(α) j

,

(α = 0,1, ...,k;y(0)i = xi ,

δδy(k)i

=∂

∂y(k)i

).

(1.4.13)Let h,v1, ...,vk be the projectors determined by (1.4.10):


h+k

∑1

vα = I ,h2 = h,vαvα = vα ,hvα = 0;

vαh= 0,vαvβ = vβ vα = 0,(α 6= β ).If we denote

XH = hX, XVα = vαX, ∀X ∈ X (TkM) (1.4.14)

we have, uniquely,

X = XH +XV1 + ...+XVk. (1.4.15)

In the adapted basis (1.4.12) we have:

XH = X(0)i δδxi ,X

V α = X(α)i δδy(α)i

,(α = 1, ...,k).

A first result on the nonlinear connectionN is as follows:

Theorem 1.4.2.The nonlinear connection N is integrable if, and onlyif:

[XH,YH ]Vα = 0, ∀X,Y ∈ χ(TkM), (α = 1, · · ·k).

1.5 The dual coefficients of a nonlinear connection

Consider a nonlinear connectionN, with the coefficients(Nij

(1)

, ...,Nij

(k)

).

The dual basis, of the adapted basis (1.4.12) is of the form

(δxi ,δy(1)i , · · · ,δy(k)i) (1.5.1)

where

δxi = dxi ,

δy(1)i = dy(1)i +Mij

(1)

dxj ,

.............................................δy(k)i = dy(k)i +Mi

j(1)

dy(k−1) j + ...+ Mij

(k−1)

dy(1) j +Mij

(k)

dxj ,

(1.5.2)

1.5 The dual coefficients of a nonlinear connection 121

and where

Mij

(1)

= Nij

(1)

, Mij

(2)

= Nij

(2)

+Nmj

(1)

Mim

(1), ....,

Mij

(k)

= Nij

(k)

+ Nmj

(k−1)

Mim

(1)+ ....+Nm

j(1)

Mim

(k−1).

(1.5.3)

The system of functions (Mij

(1)

, ...,Mij

(k)

) is called the system of dual

coefficients of the nonlinear connectionN. If the dual coefficients ofNare given, then we uniquely obtain from (1.5.3) theprimal coefficients(Ni

j(1)

, ...,Nij

(k)

,) of N.

With respect to (4.1.4), Ch. 4 the dual coefficients ofN are trans-formed by the rule

Mmj

(1)

∂ xi

∂xm = Mim

(1)

∂ xm

∂x j +∂ y(1)i

∂x j ,

.............................................

Mmj

(k)

∂ xi

∂xm = Mim

(k)

∂ xm

∂x j + Mim

(k−1)

∂ y(1)m

∂x j + · · ·+ Mim

(1)

∂ y(k−1)m

∂x j +∂ y(k)i

∂x j .

(1.5.4)These transformations of the dual coefficients characterize the non-

linear connectionN. This property allows to prove an important result:

Theorem 1.5.1 (R. Miron). For any k−semispray S with the coeffi-cients Gi the following system of functions

Mij

(1)

=∂Gi

∂y(k) j,Mi

j(2)

=12(SMi

j(1)

+Mim

(1)Mm

j(1)

), · · · ,

Mij

(k)

=1k(S Mi

j(k−1)

+Mim

(1)Mm

j(k−1)

)

(1.5.5)

gives the system of dual coefficients of a nonlinear connection, whichdepend on the k−semispray S, only.

Remark 1.5.1.Gh. Atanasiu [168] has a real contribution in demon-stration of this theorem fork= 2.

As an application we can prove:


Theorem 1.5.2.1) In the adapted basis (4.4.12) Ch. 4, the Liouville

vector fields1

Γ , ...,k

Γ can be expressed in the form

1Γ= z(1)i

δδy(k)i

,2

Γ= z(1)iδ

δy(k−1)i+2z(2)i

δδy(k)i

,

.................................................................k

Γ= z(1)iδ

δy(1)i+2z(2)i

δδy(2)i

+ · · ·+kz(k)iδ

δy(k)i,

(1.5.6)

where

z(1)i = y(1)i , 2z(2)i = 2y(2)i +Mim

(1)y(1)m, ...,

kz(k)i = ky(k)i +(k−1)Mim

(1)y(k−1)m+ · · ·+ Mi

m(k−1)

y(1)m(1.5.7)

2) With respect to (1.1.4) we have:

z(α)i =∂ xi

∂x j z(α) j , (α = 1, ...,k). (1.5.7’)

This is reason in which we callz(1)i , ...,z(k)i the distinguished Liou-ville vector fields(shortly,d-vector fields). These vectors are impor-tant in the geometry of the manifoldTkM.

A field of 1-formsω ∈ X ∗(TkM) can be uniquely written as

ω = ωH +ωV1 + · · ·+ωVk

whereωH = ω h,ωVα = ω vα , (α = 1, ...,k).

For any functionf ∈ F (TkM), the 1-formd f is

d f = (d f)H +(d f)V1 + · · ·+(d f)Vk.

In the adapted cobasis we have:

(d f)H =δ fδxi dxi ,(d f)Vα =

δ f

δy(α)iδy(α)i , (α = 1, ...,k). (1.5.8)

Let γ : I → TkM be a parametrized curve, locally expressed by

1.6 Prolongation to the manifoldTkM of the Riemannian structures 123

xi = xi(t), y(α)i = y(α)i(t), (t ∈ I), (α = 1, ...,k).

The tangent vector fielddγdt

is given by:

dγdt

=

(dγdt

)H

+

(dγdt

)V1

+ · · ·+

(dγdt

)Vk

=

=dxi

dtδ

δxi +δy(1)i

dtδ

δy(1)i+ · · ·+

δy(k)i

dtδ

δy(k)i.

The curveγ is called horizontal ifdγdt

=

(dγdt

)H

. It is characterized

by the system of differential equations

xi = xi(t),δy(1)i

dt= 0, ...,

δy(k)i

dt= 0. (1.5.9)

A horizontal curveγ is calledautoparallel curve of the nonlinearconnection ifγ = c, wherec is the extension of a curvec : I → M.

Theautoparallelcurves of the nonlinear connectionN are charac-terized by the system of differential equations

δy(1)i

dt= 0, · · · ,

δy(k)i

dt= 0,

y(1)i =dxi

dt, · · · ,y(k)i =

1k!

dkxi

dtk.

(1.5.9’)

1.6 Prolongation to the manifoldTkM of theRiemannian structures given on the base manifoldM

Applying the previous theory of the notion of nonlinear connection onthe total space of acceleration bundleTkM we can solve the old prob-lem of the prolongation of the Riemann (or pseudo Riemann) structureg given on the base manifoldM. This problem was formulated by L.Bianchi and was studied by several remarkable mathematicians as: E.Bompiani, Ch. Ehresmann, A. Morimoto, S. Kobayashi. But thesolu-


tion of this problem, as well as the solution of the prolongation toTkMof the Finsler or Lagrange structures were been recently given by R.Miron [172]. We will expound it here with very few demonstrations.

Let Rn = (M,g) be a Riemann space,g being a Riemannian metricdefined on the base manifoldM, having the local coordinategi j (x),x∈U ⊂ M. We extendgi j to π−1(U)⊂ TkM, setting

(gi j πk)(u) = gi j (x), ∀u∈ π−1(U),πk(u) = x.

gi j πk will be denoted bygi j .The problems of prolongation of the Riemannian structureg toTkM

can be formulated as follows:The Riemannian structureg on the manifoldM being a priori given,

determine a Riemannian structureG on TkM so thatG be providedonly by structureg.

As usually, we denoted byγ ijk(x) the Christoffel symbols ofg and

prove, according theorem 1.5.1:

Theorem 1.6.1.There exists nonlinear connections N on the manifold

TkM determined only by the given Riemannian structure g(x). One ofthem has the following dual coefficients

Mij

(1)

= γ ijm(x)y

(1)m,

Mij

(2)

=12

Γ Mi

j(1)

+Mim

(1)Mm

j(1)

,

.........................................

Mij

(k)

=1k

Γ Mi

j(k−1)

+Mim

(1)Mm

j(k−1)

,

(1.6.1)

whereΓ is the operator(1.2.4), Ch. 4.

Remark 1.6.1.Γ can be substituted with anyk−semisprayS, sinceMi

j(1)

, ..., Mij

(k−1)

do not depend on the variablesyk.

One proves, also:N is integrable if and only if the Riemann spaceRn = (M,g) is locally flat.

1.7 N−linear connections onTkM 125

Let us consider the adapted cobasis(δxi , δy(1)i , ..., δy(k)i) (1.5.2)to the nonlinear connectionN and to the vertical distributionsN

(1), ...,

N(k−1)

, Vk. It depend on the dual coefficients (1.6.1). So it depend on

the structureg(x) only.Now, onTkM consider the followinglift of g(x):

G= gi j (x)dxi ⊗dxj +gi j (x)δy(1)i ⊗δy(1) j+

+gi j (x)δy(k)i ⊗δy(k) j(1.6.2)

Finally, we obtain:

Theorem 1.6.2.The pair ProlkRn = (TkM,G) is a Riemann space ofdimension(k+1)n, whose metric G,(1.6.2) depends on the a priorigiven Riemann structure g(x), only.

The announced problem is solved.Some remarks:

1 Thed−Liouville vector fieldsz(1)i , · · · ,z(k)i , from (1.5.7) are con-structed only by means of the Riemannian structureg;

2 The following function

L(x,y(1), ...,y(k)) = gi j (x)z(k)iz(k) j (1.6.3)

is a regular Lagrangian which depend only on the Riemann struc-turegi j (x).

3 The previous theory, for pseudo-Riemann structuregi j (x), holds.

1.7 N−linear connections onTkM

The notion ofN−linear connection on the manifoldTkM can be stud-ied as a natural extension of that ofN−linear connection onTM, givenin the section 1.4.

Let N be a nonlinear connection onTkM having the primal coeffi-cientsNi

j(1)

, · · · ,Nij

(k)

and the dual coefficientsMij

(1)

, · · · ,Mij

(k)

.

Definition 1.7.1.A linear connectionD on the manifoldTkM is calleddistinguished ifD preserves by parallelism the horizontal distributionN. It is anN−connection if has the following property, too:


DJ = 0. (1.7.1)

We have:

Theorem 1.7.1.A linear connection D on TkM is an N-linear con-nection if and only if

(DXYH

)Vα = 0, (α = 1, · · · ,k), (DXYVα )H = 0;

(DXYVα )Vβ = 0 (α 6= β )

DX(JYH) = JDXYH ; DX(JVα) = JDXVα .

(1.7.2)

Of course, for anyN−linear connectionD we have

Dh= 0, Dvα = 0, (α = 1, · · · ,k).

SinceDXY = DXHY+DXV1Y+ · · ·+DXVkY,

settingDH

X = DXH ,DVαX = DXVα , (α = 1, · · · ,k),

we can write:

DXY = DHXY+DV1

X Y+ · · ·+DVkX Y. (1.7.3)

The operatorsDH ,DVα are not the covariant derivations but theyhave similar properties with the covariant derivations. The notion ofd−tensor fields (d−means “distinguished”) can be introduced andstudied exactly as in the section 1.3.

In the adapted basis (1.4.12) and in adapted cobasis (1.5.1)we rep-resent ad−tensor field of type(r,s) in the form

T = T i1...irj1... js

δδxi1

⊗·· ·⊗δ

δy(k)ir⊗dxj1 ⊗·· ·⊗δy(k) js. (1.7.4)

A transformation of coordinates (1.1.5), Ch. 4, has as effect thefollowing rule of transformation:

T i1...irj1... js

=∂ xi1

∂xh1· · ·

∂ xir

∂xhr

∂xk1

∂x j1· · ·

∂xks

∂x jsTh1...hr

k1...ks. (1.7.3’)


So,

1,

δδxi , · · · ,

δδy(k)i

generate the tensor algebra ofd−tensor

fields.The theory ofN−linear connection described in the chapter 1 for

casek= 1 can be extended step by step for theN−linear connectionon the manifoldTkM.

In the adapted basis (1.4.11) anN−linear connectionD has the fol-lowing form:

D δδxj

δδxi =Lm

i jδ

δxm, D δδxj

δδy(α)i

=Lmi j

δδy(α)m

,(α = 1, · · · ,k),

D δδy(β) j

δδxi =Cm

i j(β )

δδxm, D δ

δy(β) j

δδy(α)i

=Cmi j

(β )

δδy(α)m

,

(α,β = 1, · · · ,k).(1.7.5)

The system of functions

DΓ (N) = (Lmi j ,C

mi j

(1)

, ...,Cmi j

(k)

) (1.7.6)

representsthe coefficientsof D.With respect to (1.1.5),Lm

i j are transformed by the same rule as thecoefficients of a linear connection defined on the base manifold M.Others coefficientsCh

i j(α)

, (α = 1, ..,k) are transformed liked-tensors

of type(1,2).If T is ad-tensor field of type(r,s), given by (1.7.4) andX = XH =

Xi δδxi , then, by means of (1.7.5),DH

X T is:

DHX T = XmT i1...ir

j1... js|m

δδxi1

⊗·· ·⊗δ

δy(k)ir⊗dxj1 ⊗·· ·⊗δy(k) js, (1.7.7)

where

T i1...irj1... js|m

=δT i1...ir

j1... js

δxm +Li1hmThi2...ir

j1... js+ · · ·−Lh

jsmT i1...irj1...h

. (1.7.8)

The operator “|” will be called the h-covariant derivative.


Consider thevα -covariant derivativesDVαX , for X =

(α)

Xi δδy(α)i

, (α =

1, ..., k). Then, (1.7.3) and (1.7.5) lead to:

DVαX T =

(α)

Xm T i1...irj1... js

(α)

| mδ

δxi1⊗·· ·⊗

δδy(k)ir

⊗dxj1⊗·· ·⊗dxjs, (1.7.9)

where

T i1...irj1... js

(α)

| m=δT i1...ir

j1... js

δy(α)m+Ci1

hm(α)

Thi2...irj1... js

+ · · ·−Chjsm(α)

T i1...irj1... js−1h, (α = 1, ..,k).

The operators “(α)

| ”, in number ofk are calledvα -covariant deriva-tives.

Each of operators “|” and “(α)

| ” has the usual properties with respectto sum ofd-tensor or them tensor product.

Now, in the adapted basis (1.4.11) we can determine the torsionT and curvatureR of an N−linear connectionD, follows the samemethod as in the casek= 1.

We remark the following ofd−tensors of torsion:

T ijk = Li

jk −Lik j, Si

jk(α)

=Cijk

(α)

−Cik j

(α)

= 0, (α = 1, ..,k) (1.7.10)

andd−tensors of curvature

R ih jm, P

(α)

i

h jm

, S(β α)

i

h jm

, (α,β = 1, · · · ,k) (1.7.11)

The 1-forms connection of theN−linear connectionD are:

ω ij = Li

jhdxh+Cijh

(1)

δy(1)h+ · · ·+Cijh

(k)

δy(k)h. (1.7.12)

The following important theorem holds:

Theorem 1.7.2.The structure equations of an N−linear connectionD on the manifold TkM are given by:


d(dxi)−dxm∧ω im =−

(0)

Ω i

d(δy(α)i)−δy(α)m∧ω im =−

(α)

Ω i ,

dω ij −ωm

j ∧ω im =−Ω i

j ,

(1.7.13)

where(0)

Ω i ,(α)

Ω i are the2-forms of torsion and whereΩ ij are the 2-forms

of curvature:

Ω ij =

12

R ij pqdxp∧dxq+

+k

∑α=1

P ij pq(α)

dxp∧δy(α)q+k

∑α ,β=1

S ij pq

(αβ )δy(α)p∧δy(β )q.

Now, the Bianchi identities ofD can be derived from (1.7.13).The nonlinear connectionN and theN−linear connectionD allow

to study the geometrical properties of the manifoldTkM equippedwith these two geometrical object fields.

Chapter 2Lagrange Spaces of Higher–order

The concept of higher - order Lagrange space was introduced andstudied by the author of the present monograph, [155], [161].

A Lagrange space of orderk is defined as a pairL(k)n = (M,L)whereL : TkM → R is a differentiable regular Lagrangian having thefundamental tensor of constant signature. Applying the variationalproblem to the integral of action ofL we determine: a canonicalk−semispray, a canonical nonlinear connection and a canonical met-rical connection. All these are basic for the geometry of spaceL(k)n.

2.1 The spacesL(k)n = (M,L)

Definition 2.1.1.A Lagrange space of orderk ≥ 1 is a pairL(k)m =(M,L) formed by a realn−dimensional manifoldM and a differen-tiable LagrangianL : (x,y(1), · · · ,y(k)) ∈ TkM → L(x,y(1), · · · ,y(k)) ∈R for which the Hessian with the elements:

gi j =12

∂ 2L

∂y(k)i∂y(k) j(2.1.1)

has the property

rank(gi j ) = n on TkM (2.1.2)

and thed−tensorgi j has a constant signature.

Of course, we can prove thatgi j from (2.1.1) is ad−tensor field, oftype (0,2), symmetric. It is called thefundamental(or metric) tensorof the spaceL(k)n, while L is called itsfundamental function.

131

132 2 Lagrange Spaces of Higher–order

The geometry of the manifoldTkM equipped withL(x,y(1), · · · ,y(k))is called the geometry of the spaceL(k)n. We shall study this geometryusing the theory from the last chapter. Consequently, starting from the

integral of actionI(c) =∫ 1

0L

(x,

dxdt

, · · · ,1k!

dkx

dtk

)dt we determine

the Euler - Lagrange equations0Ei (L) = 0 and the Craig - Synge cov-

ectors1Ei (L), · · · ,

kEi (L). According to (3.1.14) one remarks that we

have:

Theorem 2.1.1.The equations gi jk−1E i (L) = 0 determine a k−semi-

spray

S= y(1)i∂

∂xi +2y(2)i∂

∂y(1)i+ · · ·+ky(k)i

∂∂y(k−1)i

− (k+1)Gi ∂∂y(k)i(2.1.3)

where the coefficients Gi are given by

(k+1)Gi =12

gi j

Γ(

∂∂y(k)i

)−

∂∂y(k−1)i

(2.1.4)

Γ being the operator (1.2.4).

The semispraySdepend on the fundamental functionL, only. S is

called canonical. It is globally defined on the manifoldTkM.Taking into account Theorem 1.5.1, we have:

Theorem 2.1.2.The systems of functions

Mij

(1)

=∂Gi

∂y(k) j, Mi

j(2)

=12

SMi

j(1)

+Mim

(1)Mm

j(1)

, ...,

Mij

(k)

=1k

S Mi

j(k−1)

+Mim

(1)Mm

j(k−1)

(2.1.5)

are the dual coefficients of a nonlinear connection N determined onlyon the fundamental function L of the space L(k)n.

N is the canonical nonlinear connection ofL(k)n.

2.2 Examples of spacesL(k)n 133

The adapted basis

δ

δxi ,δ

δy(1)i, · · · ,

δδy(k)i

has its dualδxi ,δy(1)i ,

..., δy(k)i. They are constructed by the canonical nonlinear connec-tion. So, the horizontal curves are characterized by the system of dif-ferential equations Part 2, Ch. 2, and the autoparallel curves ofN aregiven by Part II, Ch. 1.

The condition thatN be integrable is expressed by

[δ

δxi ,δ

δx j

]Vα

=

0, (α = 1, · · · ,k).

2.2 Examples of spacesL(k)n

1Let us consider the Lagrangian:

L(x,y(1), ...,y(k)) = gi j (x)z(k)iz(k) j (2.2.1)

wheregi j (x) is a Riemannian (or pseudo Riemannian) metric on thebase manifoldM and thez(k)i is the Liuovilled−vector field:

kz(k)i = ky(k)i +(k−1)Mim

(1)y(k−1)m+ · · ·+k Mi

m(k−1)

y(1)m (2.2.2)

constructed by means of the dual coefficients (Part II, Ch. 1)of thecanonical nonlinear connectionN from the problem of prolongationto TkM of gi j (x). So that the Lagrangian (2.2.1) depend ongi j (x)only.The pairL(k)n = (M,L), (2.2.1) is a Lagrange space of orderk. Itsfundamental tensor isgi j (x), since thed−vectorz(k)i is linearly inthe variablesy(k).

2LetL (x,y(1)) be the Lagrangian from electrodynamics

L (x,y(1)) = mcγ i j (x)y

(1)iy(1) j +2em

bi(x)y(1)i (2.2.3)

Let N be the nonlinear connection given by the theorem Part II, Ch.1, from the problem of prolongations toTkM of the Riemannian (orpseudo Riemannian) structureγ i j (x) and the Liouville tensorz(k)i

constructed by means ofN. Then the pairL(k)n = (M,L), with


L(x,y(1), ...,y(k)) = mcγ i j (x)z(k)iz(k) j +

2em

bi(x)z(k)i (2.2.4)

is a Lagrange space of orderk. It is the prolongation to the manifold

TkM of the LagrangianL (2.2.3) of electrodynamics.

These examples prove the existence of the Lagrange spaces oforderk.

2.3 Canonical metricalN−connection

Consider the canonical nonlinear connectionN of a Lagrange spaceof orderk, L(k)n = (M,L).

An N−linear connectionDwith the coefficientsDΓ (N)= (Lijk,C

ijh

(1)

,

· · · ,Cijh

(k)

) is called metrical with respect to metric tensorgi j if

gi j |h = 0, gi j

(α)

| h= 0, (α = 1, · · · ,k). (2.3.1)

Now we can prove the following theorem:


1) There exists a unique N-linear connection D onTkM verifyingthe axioms:

1 N− is the canonical nonlinear connection of space L(k)n.2 gi j |h = 0, (D is h−metrical)

3 gi j

(α)

| h= 0, (α = 1, · · · ,k), (D is vα−metrical)4 T i

jh = 0, (D is h−torsion free)

5 Sijh

(α)

= 0, (α = 1, · · · ,k), (D is vα−torsion free).

2) The coefficients CΓ (N) = (Lhi j ,C

hi j

(1)

, ...,Chi j

(k)

) of D are given by

the generalized Christoffel symbols:

2.4 The Riemannian(k−1)n−contact model of the spaceL(k)n 135

Lhi j =

12

ghs(

δgis

δx j +δgs j

δxi −δgi j

δxs

),

Chi j

(α)

=12

ghs(

δgis

δy(α) j+

δgs j

δy(α)i−

δgi j

δy(α)s

), (α = 1, ...,k).

(2.3.2)

3) D depends only on the fundamental function L of the space L(k)n.

The connectionD from the previous theorem is calledcanoni-cal metrical N-connectionand its coefficients (2.3.2) are denoted byCΓ (N).

Now, the geometry of the Lagrange spacesL(k)n can be developedby means of these two canonical connectionN andD.

2.4 The Riemannian(k−1)n−contact model of thespaceL(k)n

The almost Kahlerian model of the Lagrange spacesLn expound inthe section 7, Ch. 2, can be extended in a corresponding modelofthe higher order Lagrange spaces. But now, it is a Riemannianalmost

(k−1)n−contact structure on the manifoldTkM.The canonical nonlinear connectionN of the spaceL(k)n = (M,L)

determines the followingF (TkM)−linear mappingF : X (TkM)→

X (TkM) defined on the adapted basis toN and toNα , by

F

(δ

δxi

)=−

∂∂y(k)i

F

(δ

δy(α)i

)= 0, (α = 1, · · · ,k−1)

F

(∂

∂y(k)i

)=

δδxi , (i = 1, · · · ,n)

(2.4.1)

We can prove:

Theorem 2.4.1.We have:

1 F is globally defined onTkM.


2 F is a tensor field of type(1,1) on TkM3 KerF= N1⊕N2⊕·· ·⊕Nk−1, ImF= N0⊕Vk4 rank‖F‖= 2n5 F3+F= 0.

ThusF is an almost(k−1)n−contact structure onTkM determinedby N.

Let

(ξ1a, ξ2a, ..., ξ a

(k−1)a

), (a= 1, ...,n) be a local basis adapted to the

direct decompositionN1⊕·· ·⊕Nk−1 and

(1aη ,

2aη , ...,

(k−1)aη)

, its dual.

Thus the set(F, ξ

1a, ..., ξ

(k−1)a,1aη , ...,

(k−1)aη

), (a= 1, · · · ,n−1) (2.4.2)

is a(k−1)n almost contact structure.Indeed, (2.4.1) imply:

F(ξαa) = 0,

αaη (ξ

βb) =

δ a

b for α = β0 for α 6= β ,(α,β = 1, · · · ,(k−1)).

F2(X) =−X+n

∑a=1

k−1

∑α=1

αaη (X)ξ

αa, ∀X ∈ X (TkM),

αaη F= 0.

Let NF be the Nijenhuis tensor of the structureF.

NF(X,Y) = [FX,FY]+F2[X,Y]−F[FX,Y]−F[X,FY].

The structure (2.4.2) is said to be normal if:

NF(X,Y)+n

∑a=1

k−1

∑α=1

dαaη (X,Y) = 0, ∀X,Y ∈ X (TkM).

So we obtain a characterization of the normal structureF given by thefollowing theorem:

2.5 The generalized Lagrange spaces of orderk 137

Theorem 2.4.2.The almost(k−1)n−contact structure (2.4.2) is nor-

mal if and only if for any X,Y ∈ X (TkM) we have:

NF(X,Y)+n

∑a=1

k−1

∑α=1

d(δy(α)a) (X,Y) = 0.

The lift of fundamental tensorgi j of the spaceL(k)n with respect toN is defined by:

G=gi j dxi⊗dxj +gi j δy(1)i⊗δ y(1) j +· · ·+gi j δy(k)i⊗δy(k) j . (2.4.3)

Evidently, G is a pseudo-Riemannian structure on the manifold

TkM, determined only by spaceL(k)n.Now, it is not difficult to prove:

Theorem 2.4.3.The pair(G,F) is a Riemannian(k−1)n-almost con-

tact structure onTkM.

In this case, the next condition holds:

G(FX,Y) =−G(FY,X), ∀X,Y ∈ X (TkM).

Therefore the triple(TkM,G,F) is an metrical(k−1)n-almost contactspace named thegeometrical model of the Lagrange space of order k,L(k)n.

Using this space we can study the electromagnetic and gravitationalfields in the spacesL(k)n, [161].

2.5 The generalized Lagrange spaces of orderk

The notion of generalized Lagrange space of higher order is anaturalextension of that studied in chapter 2.

Definition 2.5.1.A generalized Lagrange space of orderk is a pairGL(k)n = (M,gi j ) formed by a real differentiablen−dimensional man-ifold M anda C∞−covariant of type (0,2), symmetricd−tensor field

gi j on TkM, having the properties:

a.gi j has a constant signature onTkM;


b. rank(gi j ) = n on TkM.

gi j is called thefundamental tensorof GL(k)n.

Evidently, any Lagrange space of orderk, L(k)n =(M,L) determinesa spaceGL(k)n with fundamental tensor

gi j =12

∂ 2L

∂y(k)i∂y(k) j. (2.5.1)

But not and conversely. Ifgi j (x,y(1), ...,y(k)) is a priori given, it ispossible that the system of differential partial equations(2.5.1) doesnot admit any solution inL(x,y(1), ...,y(k)). A necessary condition thatthe system (2.5.1) admits solutions in the functionL is thatd-tensorfield

C(k) i jh

=12

∂gi j

∂y(k)h(2.5.2)

be completely symmetric.If the system (2.5.1) has solutions, with respect toL we say that

the spaceGL(k)n is reducible to a Lagrange space of orderk. If thisproperty is not true, thenGLn is said to be nonreducible to a LagrangespaceL(k)n.

Examples

1 Let R = (M,γ i j (x)) be a Riemannian space andσ ∈ F (TkM).Consider thed-tensor field:

gi j = e2σ (γ i j πk). (2.5.3)

If∂σ

∂y(k)his a nonvanishesd−covector on the manifoldTkM, then

the pairGL(k)n = (M,gi j ) is a generalized Lagrange space of orderk and it is not reducible to a Lagrange spaceL(k)n.

2 Let Rn = (M,γ i j (x)) be a Riemann space and ProlkRn be its pro-

longation of orderk to TkM.Consider the Liouvilled−vector fieldz(k)i of ProlkRn. It is ex-pressed in the formula (2.2.2). We can introduce thed−covector

field z(k)i = γ i j z(k) j .

We assume that exists a functionn(x,y(1), · · · ,y(k))≥ 1 onTkM.

2.5 The generalized Lagrange spaces of orderk 139

Thus

gi j = γ i j +

(1−

1n2

)z(k)i z(k)j (2.5.4)

is the fundamental tensor of a spaceGL(k)n. Evidently this space isnot reducible to a spaceL(k)n, if the functionn 6= 1.

These two examples prove the existence of the generalized La-grange space of orderk.

In the last example,k = 1 leads to the metric Part I, Ch. 2 of theRelativistic Optics, (n being the refractive index).

In a generalized Lagrange spaceGL(k)n is difficult to find a nonlin-ear connectionN derived only by the fundamental tensorgi j . There-fore, assuming thatN is a priori given , we shall study the pair(N,GL(k)n). Thus of theorem of the existence and uniqueness metricalN−linear connection holds:


1 There exists an unique N-linear connection D for which

gi j |h = 0, gi j

(α)

| h= 0, (α = 1, ...,k),

T ijk = 0, Si

jk(α)

= 0, (α = 1, ...,k).

2 The coefficients of D are given by the generalized Christoffel sym-bols Part II, Ch. 2.

3 D depends on gi j and N only.

Using this theorem it is not difficult to study the geometry ofGen-eralized Lagrange spaces.

Chapter 3Higher-Order Finsler spaces

The notion of Finsler spaces of orderk, introduced by the author ofthis monograph and presented in the bookThe Geometry of Higher-Order Finsler Spaces, Hadronic Press, 1998, is a natural extension to

the manifoldTkM of the theory of Finsler spaces given in the Part I,Ch. 3. A substantial contribution in the studying of these spaces haveH. Shimada and S. Sabau [223].

The impact of this geometry in Differential Geometry, VariationalCalculus, Analytical Mechanics and Theoretical Physics isdecisive.Finsler spaces play a role in applications to Biology, Engineering,Physics or Optimal Control. Also, the introduction of the notion ofFinsler space of orderk is demanded by the solution of problem ofprolongation toTkM of the Riemannian or Finslerian structures de-fined on the base manifoldM.

3.1 Notion of Finsler space of orderk

In order to introduce the Finsler space of orderk are necessary someconsiderations on the concept of homogeneity of functions on themanifoldTkM, [161].

A function f : TkM → R of C∞−class onTkM and continuous onthe null section ofπk : TkM → M is called homogeneous of degreer ∈ Z on the fibres ofTkM (briefly r−homogeneous) if for anya∈ R+

we have

f (x,ay(1),a2y(2), · · · ,aky(k)) = ar f (x,y(1), · · · ,y(k)).

An Euler Theorem hold:

141

142 3 Higher-Order Finsler spaces

A function f∈ F (TkM), differentiable onTkM and continuous onthe null section ofπk is r−homogeneous if and only if

L kΓ

f = r f , (3.1.1)

L kΓ

being the Lie derivative with respect to the Lioville vectorfieldk

Γ .

A vector fieldX ∈ X (TkM) is r−homogeneous if

L kΓ

X = (r −1)X (3.1.1’)

Definition 3.1.1.A Finsler space of orderk, k ≥ 1, is a pairF(k)n =(M,F) determined by a real differentiable manifoldM of dimensionn and a functionF : TkM → R having the following properties:

1 F is differentiable onTkM and continuous on the null section onπk.

2 F is positive.3 F is k-homogeneous on the fibres of the bundleTkM.4 The Hessian ofF2 with the elements:

gi j =12

∂ 2F2

∂y(k)i∂y(k) j(3.1.2)

is positively defined onTkM.

From this definition it follows thatthe fundamental tensor gi j isnonsingular and 0-homogeneous on the fibres ofTkM.

Also, we remark: Any Finlser spaceF(k)n can be considered as aLagrange spaceL(k)n = (M,L), whose fundamental functionL is F2.

By means of the solution of the problem of prolongation of a FinslerstructureF(x,y(1)) toTkM we can construct some important examplesof spacesF(k)n.

A Finsler space with the propertygi j depend only on the pointsx∈ M is called a Riemann space of orderk and denoted byR(k)n.

Consequently, we have the following sequence of inclusions, simi-lar with that from Ch. 3:

R

(k)n⊂

F(k)n⊂

L(k)n⊂

GL(k)n. (3.1.3)

3.1 Notion of Finsler space of orderk 143

So, the Lagrange geometry of orderk is the geometrical theory ofthe sequence (1.1.3).

Of course the geometry ofF(k)n can be studied as the geometryof Lagrange space of orderk, L(k)n = (M,F2). Thus the canonicalnonlinear connectionN is the Cartan nonlinear connection of F(k)nand the metricalN−linear connectionD is the CartanN−metricalconnection of the spaceF(k)n, [161].

Chapter 4The Geometry ofk−cotangent bundle

4.1 Notion ofk−cotangent bundle,T∗kM

Thek−cotangent bundle(T∗kM,π∗k,M) is a natural extension of thatof cotangent bundle(T∗M,π∗,M). It is basic for the Hamilton spacesof orderk. The manifoldT∗kM must have some important properties:

1 T∗1M = T∗M;2 dimT∗kM =dimTkM = (k+1)n;3 T∗kM carries a natural Poisson structure;4 T∗kM is local diffeomorphic toTkM.

These properties are satisfied by considering the differentiable bundle(T∗kM,π∗k,M) as the fibered bundle(Tk−1MxMT∗M,πk−1xMπ∗,M).So we have

T∗kM = Tk−1MxMT∗M, π∗k = πk−1xMπ∗. (4.1.1)

A point u ∈ T∗kM is of the formu = (x,y(1), ...,y(k−1), p). It is de-

termined by the pointx = (xi) ∈ M, the accelerationy(1)i =dxi

dt, ...,

y(k−1)i =1

(k−1)!dk−1xi

dtk−1 and the momentap= (pi). The geometries

of the manifoldsTkM andT∗kM are dual via Legendre transforma-tion. Then,(xi ,y(1)i , ...,y(k−1)i, pi) are the local coordinates of a pointu∈ T∗kM.

The change of local coordinates onT∗kM is:

145

146 4 The Geometry ofk−cotangent bundle

xi = xi(x1, ...,xn), det

(∂ xi

∂x j

)6= 0,

y(1)i =∂ xi

∂x j y(1) j ,

...........

(k−1)y(k−1)i =∂ y(k−1)i

∂x j y(1) j + ...+(k−1)∂ y(k−2)i

∂y(k−2) jy(k−1) j

pi =∂x j

∂ xi p j

(4.1.2)

where the following equalities hold

∂ y(α)i

∂x j =∂ y(α+1)i

∂y(1) j= ...=

∂ y(k−1)i

∂y(k−1−α) j; (α = 0, ...,k−2;y(0) = x).

(4.1.3)The Jacobian matrixJk of (4.1.2) have the property

detJk(u) =

[det

(∂ x j

∂xi (u)

)]k−1

. (4.1.4)

The vector fields∂ i = (∂ 1, .., ∂ n) =∂

∂ pigenerate a vertical distribu-

tion Wk, while

∂

∂y(k−1)i

determine a vertical distributionVk−1, ...,

∂

∂y(1)i, ...,

∂∂y(k−1)i

determine a vertical distributionV1. We have

the sequence of inclusions:

Vk−1 ⊂Vk−2 ⊂ ...⊂V1 ⊂V,

Vu =V1,u⊕Wk,u, ∀u∈ T∗kM.

We obtain without difficulties:

Theorem 4.1.1.1 The following operators in the algebra of functionson T∗kM are the independent vector field on T∗kM:

4.1 Notion ofk−cotangent bundle,T∗kM 147

1Γ = y(1)i

∂∂y(k−1)i

2Γ = y(1)i

∂∂y(k−2)i

+2y(2)i∂

∂y(k−1)i

...........k−1Γ = y(1)i

∂∂y(1)i

+ ...+(k−1)y(k−1)i ∂∂y(k−1)i

C∗ = pi ∂ i .

(4.1.5)

2 The functionϕ = piy

(1)i (4.1.6)

is a scalar function on T∗kM.1Γ , ...,

k−1Γ are called theLiouville vector fields.

Theorem 4.1.2.1 For any differentiable function H: T∗kM → R,

T∗kM = T∗kM \0, d0H, ...,dk−2H defined by

d0H =∂H

∂y(k−1)idxi

d1H =∂H

∂y(k−2)idxi +

∂H

∂y(k−1)idy(1)i

...........

dk−2H =∂H

∂y(1)idxi +

∂H

∂y(2)idy(1)i + ...+

∂H

∂y(k−1)idy(k−2)i

(4.1.7)

are fields of1−forms onT∗kM.2 While

dk−1H =∂H∂xi dxi + ...+

∂H

∂y(k−1)idy(k−1)i (4.1.8)

is not a field of1−form.3 We have

dH = dk−1H + ∂ iH pi . (4.1.9)

If H = ϕ = piy(1)i , thend0H = ...= dk−1H = 0 and

148 4 The Geometry ofk−cotangent bundle

ω = dk−2ϕ = pidxi (4.1.10)

ω is called the Liouville 1−form. Its exterior differential is expressedby

θ = dω = dpi ∧dxi . (4.1.11)

θ is a 2−form of rank 2n< (k+1)n = dimT∗kM, for k > 1. Conse-quenceθ is a presymplectic structure onT∗kM.

Let us consider the tensor field of type(1,1) on T∗kM:

J=∂

∂y(1)i⊗dxi +

∂∂y(2)i

⊗dy(1)i + ...+∂

∂y(k−1)i⊗dy(k−2)i . (4.1.12)


1 J is globally defined.2 J is integrable.3 JJ ...J= Jk = 0.4 kerJ =Vk−1⊕Wk.5 rankJ = (k−1)n.

6 J(1Γ ) = 0, J(

2Γ ) =

1Γ , ..., J(

k−1Γ ) =

k−2Γ , J(C∗) = 0.

Theorem 4.1.4.1 For any vector field X∈X (T∗kM),1X, ...,

k−1X given

by1X = J(X),

2X = J2X, ...,

k−1X = Jk−1X (4.1.13)

are vector fields.

2 If X =(0)iX

∂∂xi +

(1)iX

∂∂y(1)i

+ ...+(k−1)i

X∂

∂y(k−1)i+Xi ∂ i . Then

1X = J(X) =

(0)iX

∂∂y(1)i

+ ...+(k−2)i

X∂

∂y(k−1)i, ...,

k−1X =

0X

i ∂∂y(k−1)i

.

(4.1.14)

On the manifoldT∗kM there exists aPoisson structuregiven by

f ,g0 =∂ f∂xi

∂g∂ pi

−∂ f∂ pi

∂g∂xi

f ,gα =∂ f

∂y(α)i

∂g∂ pi

−∂ f∂ pi

∂g

∂y(α)i(α = 1, ...,k−1).

(4.1.15)

4.1 Notion ofk−cotangent bundle,T∗kM 149

It is not difficult to study the properties of this structure.For details see the book [163].

Part IIIAnalytical Mechanics of Lagrangian and

Hamiltonian Mechanical Systems

This part is devoted to applications of the Lagrangian and Hamilto-nian geometries of orderk= 1 andk> 1 to Analytical Mechanics.

Firstly, to classical Mechanics of Riemannian mechanical systemsΣR = (M,T,Fe) for which the external forcesFe depend on the ma-

terial pointx∈ M and on the velocitiesdxi

dt. So, in general,ΣR are the

nonconservative systems. Then we are obliged to takeFe as a verti-cal field on the phase spaceTM and apply the Lagrange geometry forstudy the geometrical theory ofΣR . More general, we introduce thenotion on Finslerian mechanical systemΣF = (M,F,Fe), whereM isthe configuration space,TM is the velocity space,F is the fundamen-

tal function of a given Finsler spaceFn= (M,F(x,y)) andFe

(x,

dxdt

)

are the external forces defined as vertical vector field on thevelocityspaceTM. The fundamental equations ofΣF are theLagrangeequa-tions

ddt

∂F2

∂yi −∂F2

∂xi = Fi(x,y), yi =dxi

dt,

F2 being the energy of Finsler spaceFn.More general, consider a tripleΣL = (M,L,Fe), whereM is space

of configurations,Ln = (M,L) is a Lagrange space, andFeare the ex-ternal forces. The fundamental equations ate the Lagrange equations,too.

The dual theory leads to the Cartan and Hamiltonian mechanicalsystems and is based on the Hamilton equations.

Finally, we remark the extension of such kind of analytical mechan-ics, to the higher order. Some applications will be done.

These considerations are based on the paper [175] and on the papersof J. Klein [123], M. Crampin [64], Manuel de Leon [138]. Also, weuse the papers of R. Miron, M. Anastasiei, I. Bucataru [166],and ofR. Miron, H. Shimada, S. Sabau and M. Roman [174], [180].

Chapter 1Riemannian mechanical systems

1.1 Riemannian mechanical systems

Let gi j (x) be Riemannian tensor field on the configuration spaceM.So its kinetic energy is

T =12

gi j (x)yiy j , yi =

dxi

dt= xi . (1.1.1)

Following J. Klein [123], we can give:

Definition 1.1.1.A Riemannian Mechanical system (shortly RMS) isa tripleΣR = (M,T,Fe), where

1M is ann−dimensional, real, differentiable manifold (called con-figuration space).

2T =12

gi j (x)xi x j is the kinetic energy of an a priori given Rieman-

nian spaceRn = (M,gi j (x)).

3Fe(x,y) = F i(x,y)∂

∂yi is a vertical vector field on the velocity space

TM (Feare called external forces).

Of course,ΣR is a scleronomic mechanical system. The covariantcomponents ofFeare:

Fi(x,y) = gi j (x)Fi(x,y). (1.1.2)

Examples:

1. RMS - for whichFe(x,y) = a(x,y)C, a 6= 0. ThusF i = a(x,y)yi andΣR is called a Liouville RMS.

153

154 1 Riemannian mechanical systems

2. The RMSΣR , whereFe(x,y) = F i(x) ∂∂yi , andFi(x) = gradi f (x),

called conservative systems.3. The RMSΣR , whereFe(x,y) = F i(x) ∂

∂yi , but Fi(x) 6= gradi f (x),called non-conservative systems.

Remark 1.1.1.1. A conservative systemΣR is called by J. Klein [123]a Lagrangian system.

2. One should pay attention to not make confusion of this kindofmechanical systems with the “Lagrangian mechanical systems”ΣL = (M,L(x,y),Fe(x,y)) introduced by R. Miron [175], whereL : TM → R is a regular Lagrangian.

Starting from Definition 1.1.1, in a very similar manner as inthegeometrical theory of mechanical systems, one introduces

Postulate.The evolution equations of a RSMΣR are theLagrangeequations:

ddt

∂L∂yi −

∂L∂xi = Fi(x,y), yi =

dxi

dt, L = 2T. (1.1.3)

This postulate will be geometrically justified by the existence of asemispraySon TM whose integral curves are given by the equations(1.1.3). Therefore, the integral curves of Lagrange equations will becalled theevolution curvesof the RSMΣR .

The LagrangianL = 2T has the fundamental tensorgi j (x).

Remark 1.1.2.In classical Analytical Mechanics, the coordinates(xi)of a material pointx∈ M are denoted by(qi), and the velocitiesyi =dxi

dtby qi =

dqi

dt. However, we prefer to use the notations(xi) and(yi)

which are often used in the geometry of the tangent manifoldTM.

The external forcesFe(x,y) give rise to the one-form

σ = Fi(x,y)dxi . (1.1.4)

SinceFe is a vertical vector field it follows thatσ is semibasic oneform. Conversely, ifσ from (1.1.4) is semibasic one form, thenFe=F i(x,y) ∂

∂yi , with F i = gi j Fj , is a vertical vector field on the manifoldTM. J Klein introduced the the external forces by means of a one-formσ , while R. Miron [175] definedFeas a vertical vector field onTM.

The RMSΣR is a regular mechanical system because the Hessian

matrix with elements∂ 2T

∂yi∂y j = gi j (x) is nonsingular.

1.1 Riemannian mechanical systems 155

We have the following important result.

Proposition 1.1.1.The system of evolution equations(1.1.3) are equiv-alent to the following second order differential equations:

d2xi

dt2+ γ i

jk(x)dxj

dtdxk

dt=

12

F i(x,dxdt

), (1.1.5)

whereγki j (x) are the Christoffel symbols of the metric tensor gi j (x).

In general, for a RMSΣR , the system of differential equations(1.1.5) is not autoadjoint, consequently, it can not be written as theEuler-Lagrange equations for a certain Lagrangian.

In the case of conservative RMS, with12

Fi(x) = −∂U(x)

∂xi , here

U(x) a potential function, the equations (1.1.2) can be written asEuler-Lagrange equations for the LagrangianT +U . They haveT +U =constant as a prime integral.

This is the reason that the nonconservative RMSΣR , with Fe de-

pending onyi =dxi

dtcannot be studied by the methods of classical me-

chanics. A good geometrical theory of the RMSΣR should be basedon the geometry of the velocity spaceTM.

From (1.1.4) we can see that in the canonical parametrization t = s(s being the arc length in the Riemannian spaceRn), we obtain thefollowing result:

Proposition 1.1.2.If the external forces are identically zero, then theevolution curves of the systemΣR are the geodesics of the RiemannianspaceRn.

In the following we will study how the evolution equations changewhen the spaceRn = (M,g) is replaced by another Riemannian spaceRn = (M, g) such that:

1 Rn andRn have the same parallelism of directions;2 Rn andRnhave same geodesics;3 Rn is conformal toRn.

In each of these cases, the Levi-Civita connections of thesetwo Riem-manian spaces are transformed by the rule:

1 γ ijk(x) = γ i

jk(x)+δ ijαk(x);

2 γ ijk(x) = γ i

jk(x)+δ ijαk(x)+δ i

kα j(x);

3 γ ijk(x) = γ i

jk(x)+δ ijαk(x)+δ i

kα j(x)−g jk(x)α i(x),whereαk(x) is an arbitrary covector field onM, andα i(x)=gi j (x)α j(x).


It follows that the evolution equations (1.1.3) change to the evolu-tion equations of the systemΣ

Ras follows:

1 In the first case we obtain

d2xi

dt2+ γ i

jk(x)dxj

dtdxk

dt=−α

dxi

dt+

12

F i(x,dxdt

);α = αk(x)dxk

dt(1.1.6)

Therefore, even thoughΣR is a conservative system, the mechanicalsystemΣ

Ris nonconservative system having the external forces

Fe=

(−α(x,y)yi +

12

F i(x,y)

)∂

∂yi , α = αk(x)dxk

dt.

2 In the second case we have

d2xi

dt2+ γ i

jk(x)dxj

dtdxk

dt=−2α

dxi

dt+

12

F i(

x,dxdt

); α = αk(x)

dxk

dt(1.1.7)

and

Fe=

(−2α(x,y)yi +

12

F i(x,y)

)∂

∂yi , α = αk(x)dxk

dt.

3 In the third caseFe is

Fe=

2(−αyi +Tα i)+

12

F i

∂∂yi , α = αk(x)

dxk

dt,

α i(x) = gi j (x)α j(x).

The previous properties lead to examples with very interestingproperties.

1.2 Examples of Riemannian mechanical systems

Recall that in the case of classicalconservative mechanical systemswe have 2Fe= grad U , whereU (x) is a potential function. There-fore, the Lagrange equations are given by

ddt

∂∂yi (T +U )−

∂∂xi (T +U ) = 0.

1.2 Examples of Riemannian mechanical systems 157

We obtain from here a prime integralT +U = h (constant) whichgive usthe energy conservation law.

In the nonconservative case we have numerous examples suggestedby 1, 2, 3 from the previous section, where we takeF i(x,y) = 0.

Other examples of RMS can be obtained as follows

1.

2Fe=−β (x,y)yi ∂∂yi , (1.2.1)

whereβ = β i(x)yi is determined by the electromagnetic potentials

β i(x), (i = 1, ..,n).2.

Fe= (T −β )yi ∂∂yi , (1.2.2)

whereβ = β i(x)yi andT is the kinetic energy.

3. In the three-body problem, M. Barbosu [36] applied the followingconformal transformation:

ds2 = (T +U )ds2

to the classic Lagrange equations and had obtained a nonconserva-tive mechanical system with external force field

Fe= (T +U)yi ∂∂yi .

4. The external forcesFe=F i(x)∂

∂yi lead to classical nonconservative

Riemannian mechanical systems. For instance, forFi =−gradi U +Ri(x) whereRi(x) are the resistance forces, and the configurationspaceM isR3.

5. If M = R3, T =12

mδ i j yiy j andFe= 2F i(x)∂

∂yi , then the evolution

equations are

md2xi

dt2= F i(x),

which isthe Newton’s law.6. The harmonic oscillator.

M =Rn, gi j = δ i j , 2Fi =−ω2i xi (the summation convention is not

applied) andω i are positive numbers,(i = 1, ..,n).The functions


hi = (xi)2+ω2i xi , and H =

n

∑i=1

hi

are prime integrals.7. Suggested by the example 6, we consider a systemΣR with Fe=

−2ω(x)C, whereω(x) is a positive function andC is the Liouvillevector field.The evolution equations, in the caseM = Rn, are given by

d2xi

dt2+ω(x)

dxi

dt= 0.

Puttingyi =dxi

dt, we can write

dyi

dt+ω(x)yi = 0, (i = 1, ..,n).

So, we obtain yi = Cie−∫

ω(x(t))dt and thereforexi = Ci0+

Ci∫

e−∫

ω(x(t))dtdt.

8. We can consider the systemsΣR having

Fe= 2aijk(x)y

jyk ∂∂yi ,

whereaijk(x) is a symmetric tensor field onM. The external force

field Fe has homogeneous components of degree 2 with respect toyi .

9. Relativistic nonconservative mechanical systems can beobtainedfor a Minkowski metric in the space-timeR4.

10. A particular case of example 1 above [223] is the case when theexternal force field coefficientsF i(x,y) are linear inyi , i.e.

Fe= 2F i(x,y)∂

∂yi = 2Yik(x)y

k ∂∂yi ,

whereY : TM→TM is a fiber diffeomorphism calledLorentz force,namely for anyx∈ M, we have

Yx : TxM → TxM, Yx

(∂

∂xi

)=Y j

i (x)∂

∂x j .

1.3 The evolution semispray of the mechanical systemΣR 159

Let us remark that in this case, formally, we can write the Lagrangeequations of this RMS in the form

∇γ γ =Y(γ),

where∇ is the Levi-Civita connection of the Riemannian space(M,g) and γ is the tangent vector along the evolution curvesγ :[a,b]→ M.This type of RMS is important because of the global behavior of itsevolution curves.Let us denote byS the evolutionary semispray, i.e.S is a vectorfield onTM which is tangent to the canonical liftγ = (γ, γ) of theevolution curves (see the following section for a detailed discussionon the evolution semispray).We will denote byTa the energy levels of the Riemannian metricg,i.e.

Ta =

(x,y) ∈ TM : T(x,y) =

a2

2

,

whereT is the kinetic energy ofg, anda is a positive constant. Onecan easily see thatTa is the hypersurface inTM of constant Rie-mannian length vectors, namely for anyX = (x,y) ∈ Ta, we musthave|X|g = a, where|X|g is the Riemannian length of the vectorfield X onM.If we restrict ourselves for a moment to the two dimensional case,then it is known that for sufficiently small values ofc the restrictionof the flow of the semispraySto Tc contains no less than two closedcurves whenM is the 2-dimensional sphere, and at least three oth-erwise. These curves projected to the base manifoldM will giveclosed evolution curves for the given Riemannian mechanical sys-tem.

1.3 The evolution semispray of the mechanical systemΣR

Let us assume thatFe is global defined onM, and consider the me-chanical systemΣR = (M,T,Fe). We have

Theorem 1.3.1 ([175]).The following properties hold good:1 The quantity S defined by


S= yi ∂∂xi − (2

Gi −12

F i)∂

∂yi ,

2

Gi= γ ijky jyk,

(1.3.1)

is a vector field on the velocity space TM.2 S is a semispray, which depends onΣR only.3 The integral curves of the semispray S are the evolution curves

of the systemΣR .

Proof. 1 Writing S in the form

S=S+

12

Fe, (1.3.2)

whereS is the canonical semispray with the coefficients

Gi , we cansee immediately thatS is a vector field onTM.

2 SinceS is a semispray andFea vertical vector field, it followsS

is a semispray. From (1.3.1) we can see thatSdepends onΣR , only.3 The integral curves ofSare given by

dxi

dt= yi ;

dyi

dt+2

Gi (x,y) =12

F i(x,y). (1.3.3)

Replacingyi in the second equation we obtain (1.1.2)

S will be called the evolutionor canonical semisprayof the non-conservative Riemannian mechanical systemΣR . In the terminologyof J. Klein [123],S is the dynamical system ofΣR .

Based onSwe can develop the geometry of the mechanical systemΣR onTM.

Let us remark thatScan also be written as follows:


∂∂yi , (1.3.1’)

with the coefficients

2Gi = 2

Gi −12

F i. (1.3.4)

We point out thatS is homogeneous of degree 2 if and only ifF i(x,y) is 2-homogeneous with respect toyi . This property is not sat-

1.4 The nonlinear connection ofΣR 161

isfied in the case∂F i

∂y j ≡ 0, and it is satisfied for examples 1 and 7 from

section 1.2.We have

Theorem 1.3.2.The variation of the kinetic energy T of a mechanicalsystemΣR , along the evolution curves(1.1.2), is given by:

dTdt

=12

Fidxi

dt. (1.3.5)

Proof. A straightforward computation gives

dTdt

=∂T∂xi

dxi

dt+

∂T∂yi

dyi

dt=

(ddt

∂T∂yi −

12

Fi

)dxi

dt+

∂T∂yi

dyi

dt=

=ddt

(yi ∂T

∂yi

)−

12

Fidxi

dt= 2

dTdt

−12

Fidxi

dt,

and the relation (1.3.5) holds good.

Corollary 1.3.1. T =constant along the evolution curves if and onlyif the Liouville vectorC and the external force Fe are orthogonalvectors along the evolution curves ofΣ .

Corollary 1.3.2. If Fi = gradi U then ΣR is conservative and T+U = h (constant) on the evolution curves ofΣR .

If the external forcesFe are dissipative, i.e.〈C,Fe〉< 0, then fromthe previous theorem, it follows a result of Bucataru-Miron(see [49]):

Corollary 1.3.3. The kinetic energy T decreases along the evolutioncurves if and only if the external forces Fe are dissipative.

Since the energy ofΣR is T (the kinetic energy), the Theorem 1.3.2holds good in this case. The variation ofT is given by (1.3.5) andhence we obtain:T is conserved along the evolution curves ofΣR ifand only if the vector fieldFe and the Liouville vector fieldC areorthogonal.

1.4 The nonlinear connection ofΣR

Let us consider the evolution semispraySof ΣR given by


S= yi ∂∂xi − (2

G

i−

12

F i)∂

∂yi (1.4.1)

with the coefficients

2Gi = 2

Gi −12

F i. (1.4.2)

Consequently, the evolution nonlinear connectionN of the mechanicalsystemΣR has the coefficients:

Nij =

Nij −

14

∂F i

∂y j = γ ijkyk−

14

∂F i

∂y j . (1.4.3)

If the external forcesFe does not depend by velocitiesyi =dxi

dt,

thenN =N .

Let us consider thehelicoidal vector field(see Bucataru-Miron,[49], [50])

Pi j =12

(∂Fi

∂y j −∂Fj

∂yi

)(1.4.4)

and the symmetric part of tensor∂Fi

∂y j :

Qi j =12

(∂Fi

∂y j +∂Fj

∂yi

). (1.4.5)

OnTM, P gives rise to the 2-form:

(1.4.4′) P= Pi j dxi ∧dxj

andQ is the symmetric vertical tensor:

(1.4.5′) Q= Qi j dxi ⊗dxj .

Denoting by∇ the dynamical derivative with respect to the pair(S,N),one proves the theorem of Bucataru-Miron:

Theorem 1.4.1.For a Riemannian mechanical systemΣR =(M,T,Fe)the evolution nonlinear connection is the unique nonlinearconnectionthat satisfies the following conditions:

1.4 The nonlinear connection ofΣR 163

(1)∇g=−12

Q

(2)θL(hX,hY) =12

P(X,Y), ∀X,Y ∈ χ(TM),

where

θL = 4gi j

δ y j ∧dxi (1.4.6)

is the symplectic structure determined by the metric tensorgi j and the

nonlinear connectionN.

The adapted basis of the distributionsN andV is given by

(δ

δxi ,∂

∂yi

),

where

δδxi =

∂∂xi −N j

i∂

∂y j =

δδxi +

14

∂F j

∂yi

∂∂y j (1.4.7)

and its dual basis(dxi ,δyi) has the 1-formsδyi expressed by

δyi = dyi +Nijdxj =

δyi −14

∂F i

∂y j dxj . (1.4.8)

It follows that the curvature tensorR ijk of N (from (1.4.3) is

Rki j =

δNki

δx j −δNk

j

δxi =

(δ

δx j

∂∂yi −

δδxi

∂∂y j

)(

Gk −14

Fk) (1.4.9)

and the torsion tensor ofN is:

tki j =

∂Nki

∂y j −∂Nk

j

∂yi =

(∂

∂y j

∂∂yi −

∂∂yi

∂∂y j

)(

Gk −14

Fk) = 0.

(1.4.10)These formulas have the following consequences.

1. The evolution nonlinear connectionN of ΣR is integrable if andonly if the curvature tensorR i

jk vanishes.2. The nonlinear connection is torsion free, i.e.tk

i j = 0.

The autoparallel curves of the evolution nonlinear connection N aregiven by the system of differential equations


d2xi

dt2+Ni

j

(x,

dxdt

)dxj

dt= 0,

which is equivalent to

d2xi

dt2+

Nij

(x,

dxdt

)dxj

dt=

14

∂F i

∂y j

dxj

dt. (1.4.11)

In the initial conditions

(x0,

(dxdt

)

0

), locally one uniquely deter-

mines the autoparallel curves ofN.If F i is 2-homogeneous with respect toyi , then the previous system

coincides with the Lagrange equations (1.1.4).Therefore, we have:

Theorem 1.4.2.If the external forces Fe are2-homogeneous with re-

spect velocities yi =dxi

dt, then the evolution curves ofΣR coincide

to the autoparallel curves of the evolution nonlinear connection N ofΣR .

In order to proceed further, we need the exterior differential of 1-formsδyi .

One obtains

d(δyi) = dNij ∧dxj =

12

Rik jdxj ∧dxk+Bi

k jδy j ∧dxk, (1.4.12)

where

Bik j = Bi

jk =∂ 2Gi

∂yk∂y j (1.4.13)

are the coefficients of the Berwald connection determined bythe non-linear connectionN.

1.5 The canonical metrical connectionCΓ (N)

The coefficients of the canonical metrical connectionCΓ (N) =(F i

jk ,Ci

jk)

are given by the generalized Christoffel symbols [166]:

1.5 The canonical metrical connectionCΓ (N) 165

F ijk =

12

gis

(δgsk

δx j +δg js

δxk −δg jk

δxs

),

Cijk =

12

gis

(∂gsk

∂y j +∂g js

∂yk −∂g jk

∂ys

) (1.5.1)

wheregi j (x) is the metric tensor ofΣR .

On the other hand, we haveδg jk

δxi =∂g jk

∂xi and∂g jk

∂yi = 0, and there-

fore we obtain:

Theorem 1.5.1.The canonical metrical connection CΓ (N) of the me-chanical systemΣR has the coefficients

F ijk(x,y) = γ i

jk(x), Cijk(x,y) = 0. (1.5.2)

Let ω ij be the connection forms ofCΓ (N):

ω ij = F i

jkdxk+Cijkδyk = γ i

jk(x)dxk. (1.5.3)

Then, we have ([166]):

Theorem 1.5.2.The structure equation of CΓ (N) can be expressed by


1Ω i,


2Ω i,

dω ij − ωk

j ∧ω ik =− Ω i

j ,

(1.5.4)

where the2-forms of torsion1

Ω i ,2

Ω i are as follows

1Ω i =Ci

jkdxj ∧δyk = 0,

2Ω i = Ri

jkdxj ∧dxk+Pijkdxj ∧δyk.

(1.5.5)

Here Rijk is the curvature tensor of N and Pi

jk = γ ijk − γ i

k j = 0.

The curvature 2-formΩ ij is given by

Ω ij =

12

R ij kh dxk∧dxh+P i

j kh dxk∧δyh+12

S ij kh δyk∧δyh, (1.5.6)


where

R ih jk =

δF ih j

δxk −δF i

hk

δx j +Fsh jF i

sk−FshkF i

s j+CihsRs

jk =

=∂γ i

h j

∂xk −∂γ i

hk

∂x j + γsh jγ i

sk− γshkγ i

s j = r ih jk

(1.5.7)

is the Riemannian tensor of curvature of the Levi-Civita connectionγ i

jk(x) and the curvature tensorsP ij kh,S i

j kh vanish.Therefore, the tensors of torsion ofCΓ (N) are

Rijk, T i

jk = 0, Sijk = 0, Pi

jk = 0, Cijk = 0 (1.5.8)

and the curvature tensors ofCΓ (N) are

R ij kh(x,y) = r i

j kh(x), P ij kh(x,y) = 0, S i

j kh(x,y) = 0. (1.5.9)

The Bianchi identities can be obtained directly from (1.5.4), takinginto account the conditions (1.5.8) and (1.5.9).

Theh- andv-covariant derivatives ofd-tensor fields with respect toCΓ (N) = (γ i

jk,0) are expressed, for instance, by

∇kti j =δ ti jδxk − γs

ikts j− γsjktis,

∇kti j =∂ ti j∂yk −Cs

ikts j−Csjktis =

∂ ti j∂yk .

Therefore,CΓ (N) being a metric connection with respect togi j (x),we have

∇k gi j =∇kgi j = 0, (1.5.10)

(∇ is the covariant derivative with respect to Levi-Civita connection

of gi j ) and∇k gi j = 0. (1.5.10’)

The deflection tensors ofCΓ (N) are

Dij = ∇ jy

i =δyi

δx j +ysγ is j =−Ni

j +ysγ is j

1.6 The electromagnetism in the theory of the Riemannian mechanical systemsΣR 167

anddi

j = ∇ j yi = δ ij .

The evolution nonlinear connection of a Riemannian mechanicalsystemΣR given by (1.4.3) implies

Dij =−

N

i

j +14

∂F i

∂y j +ysγ is j =

14

∂F i

∂y j ,

where we have usedD j yi =

D

i

j= 0. It follows

Proposition 1.5.1.For a Riemannian mechanical system the deflec-tion tensors Di j and di

j of the connection CΓ (N) are expressed by

Dij =

14

∂F i

∂y j , dij = δ i

j . (1.5.11)

1.6 The electromagnetism in the theory of theRiemannian mechanical systemsΣR

In a Riemannian mechanical systemΣR = (M,T,Fe) whose exter-

nal forcesFe depend on the pointx and on the velocityyi =dxi

dt,

the electromagnetic phenomena appears because the deflection ten-sorsDi

j anddij from (1.5.11) nonvanish. Hence thed-tensorsDi j =

gihDhj , di j = gihδ h

j = gi j determine theh-electromagnetic tensorFi jandv-electromagnetic tensorfi j by the formulas, [166]:


12(di j −d ji ). (1.6.1)

By means of (1.5.11), we haveProposition 1.6.1.The h- and v-tensor fieldsFi j and fi j are given by

Fi j =14

Pi j , fi j = 0. (1.6.2)

where Pi j is the helicoidal tensor(1.4.4) of ΣR .

Indeed, we have


Fi j =14(Di j −D ji ) =

18

(∂F i

∂y j −∂F j

∂yi

)=

14

Pi j .

If we denoteRi jk := gihRhjk, then we can prove:

Theorem 1.6.1.The electromagnetic tensorFi j of the mechanicalsystemΣR = (M,T,Fe) satisfies the following generalized Maxwellequations:

∇kF ji +∇iFk j +∇ jFik =−(Rk ji +Rik j +Rjik). (1.6.3)

∇kF ji = 0. (1.6.3’)

Proof. Applying the Ricci identities to the Liouville vector field,weobtain

∇iDkj −∇ jD

ki = yhr k

h i j −Rki j , ∇id

kj − ∇ jD

ki = 0

and this leads to

∇iDk j −∇ jDki = yhrhki j −Rki j , (1.6.4)

∇ jDki = 0. (1.6.4’)

By taking cyclic permutations of the indices i,k,j and adding in (1.6.4),by taking into account the identityrhi jk + rh jki + rhki j = 0 we deduce(1.6.3), and analogously (1.6.3’).

From equations (1.6.2) and (1.6.4’) we obtain as consequences:

Corollary 1.6.4. The electromagnetic tensorFi j of the mechanical

systemΣR = (M,T,Fe) does not depend on the velocities yi =dxi

dt.

Indeed, by means of (1.6.3’) we have∇ jFik =∂Fik

∂y j = 0.

In other words, the helicoidal tensorPi j of ΣR does not depend on

the velocitiesyi =dxi

dt.

We end the present section with a remark: this theory has applica-tions to the mechanical systems given by example 1 in Section 1.2.

1.7 The almost Hermitian model of the RMSΣR 169

1.7 The almost Hermitian model of the RMSΣR

Let us consider a RMSΣR = (M,T(x,y),Fe(x,y)) endowed with theevolution nonlinear connection with coefficientsNi

j from (1.4.3) andwith the canonicalN-metrical connectionCΓ (N) = (γ i

jk(x),0). Thus,

on the velocity spaceTM = TM \ 0 we can determine an almostHermitian structureH2n = (TM,G,F) which depends on the RMSonly.

Let

(δ

δxi ,∂

∂yi

)be the adapted basis to the distributionsN andV

and its adapted cobasis(dxi ,δyi), where

δδxi =

δδxi +

14

∂Fs

∂yi∂

∂ys

δyi =

δ yi −14

∂F i

∂ys dxs

(1.7.1)

The lift of the fundamental tensorgi j (x) of the Riemannian spaceRn = (M,gi j (x)) is defined by

G= gi j dxi ⊗dxj +gi j δyi ⊗δy j , (1.7.2)

and the almost complex structureF, determined by the nonlinear con-nectionN, is expressed by

F=−∂

∂yi ⊗dxi +δ

δxi ⊗δyi . (1.7.3)

Thus, the following theorems hold good.


1. The pair(TM,G) is a pseudo-Riemannian space.2. The tensorG depends onΣR only.3. The distributions N and V are orthogonal with respect toG.

Theorem 1.7.2.1. The pair(TM,F) is an almost complex space.2. The almost complex structureF depends onΣR only.3.F is integrable on the manifoldTM if and only if the d-tensor field

Rijk(x,y) vanishes.


Also, it is not difficult to prove

Theorem 1.7.3.We have

1. The triple H2n = (TM,G,F) is an almost Hermitian space.2. The space H2n depends onΣR only.3. The almost symplectic structure of H2n is

θ = gi j δyi ∧dxj . (1.7.4)

If the almost symplectic structureθ is a symplectic one (i.e.dθ =0), then the spaceH2n is almost Kahlerian.

On the other hand, using the formulas (1.7.4), (1.4.12) one obtains

dθ =13!(Ri jk +Rjki +Rki j)dxi ∧dxj ∧dxk

+12(gisBs

jk −g jsBsik)δyk∧dxj ∧dxi .

Therefore, we deduce

Theorem 1.7.4.The almost Hermitian space H2n is almost Kahlerianif and only if the following relations hold good

Ri jk +Rjki +Rki j = 0, gisBsjk −g jsB

sik = 0. (1.7.5)

The spaceH2n = (TM,G,F) is called thealmost Hermitian modelof the RMSΣR .

One can use the almost Hermitian modelH2n to study the geomet-rical theory of the mechanical systemΣR . For instance the Einsteinequations of the RMSΣR are the Einstein equations of the pseudo-Riemannian space(TM,G) (cf. Chapter 6, part 1).

Remark 1.7.3.The previous theory can be applied without difficultiesto the examples 1-8 in Section 1.2.

Chapter 2Finslerian Mechanical systems

The present chapter is devoted to the Analytical Mechanics of theFinslerian Mechanical systems. These systems are defined bya tripleΣF = (M,F2,Fe) whereM is the configuration space,F(x,y) is thefundamental function of a semidefinite Finsler spaceFn=(M,F(x,y))andFe(x,y) are the external forces. Of course,F2 is the kinetic energyof the space. The fundamental equations are the Lagrange equations:

Ei(F2)≡

ddt

∂F2

∂ xi −∂F2

∂xi = Fi(x, x).

We study here the canonical semisprayS of ΣF and the geometry ofthe pair(TM,S), whereTM is velocity space.

One obtain a generalization of the theory of Riemannian Mechani-cal systems, which has numerous applications and justifies the intro-duction of such new kind of analytical mechanics.

2.1 Semidefinite Finsler spaces

Definition 2.1.1.A Finsler space with semidefinite Finsler metric is apair Fn = (M,F(x,y)) where the functionF : TM → R satisfies thefollowing axioms:

1F is differentiable onTM and continuous on the null section ofπ : TM → M;

2F ≥ 0 onTM;3F is positive 1-homogeneous with respect to velocities ˙xi = yi .4The fundamental tensorgi j (x,y)

171

172 2 Finslerian Mechanical systems

gi j =12

∂ 2F2

∂yi∂y j (2.1.1)

has a constant signature onTM;5The Hessian of fundamental functionF2 with elementsgi j (x,y) is

nonsingular:det(gi j (x,y)) 6= 0 onTM. (2.1.2)

Example. If gi j (x) is a semidefinite Riemannian metric onM, then

F =√

|gi j (x)yiy j | (2.1.3)

is a function with the propertyFn = (M,F) is a semidefinite Finslerspace.

Any Finsler spaceFn = (M,F(x,y)), in the sense of definition3.1.1, part I, is a definite Finsler space. In this case the property 5is automatical verified.

But, these two kind of Finsler spaces have a lot of common prop-erties. Therefore, we will speak in general on Finsler spaces. The fol-lowing properties hold:

1The fundamental tensorgi j (x,y) is 0-homogeneous;2F2 = gi j (x,y)yiy j ;

3 pi =12

∂F2

∂yi is d−covariant vector field;

4The Cartan tensor

Ci jk =14

∂ 3F2


∂gi j

∂yk (2.1.4)

is totally symmetric and

yiCi jk =C0 jk = 0. (2.1.5)

5ω = pidxi is 1-form onTM (the Cartan 1-form);6θ = dω = dpi ∧dxi is 2-form (the Cartan 2-form);7The Euler-Lagrange equations ofFn are

Ei(F2) =

ddt

∂F2

∂yi −∂F2

∂xi = 0, yi =dxi

dt(2.1.6)

8The energyEF of Fn is

2.1 Semidefinite Finsler spaces 173

EF = yi ∂F2

∂yi −F2 = F2 (2.1.7)

9 The energyEF is conserved along to every integral curve ofEuler-Lagrange equations (2.1.6);

10 In the canonical parametrization, the equations (2.1.6) give thegeodesics ofFn;

11 The Euler-Lagrange equations (2.1.6) can be written in the equiv-alent form

d2xi

dt2+ γ i

jk(x,dxdt

)dxj

dtdxk

dt= 0, (2.1.8)

whereγ ijk

(x,

dxdt

)are the Christoffel symbols of the fundamental

tensorgi j (x,y).12 The canonical semisprayS is


∂∂yi (2.1.9)

with the coefficients:

2Gi(x,y) = γ ijk(x,y)y

jyk = γ i00(x,y), (2.1.9’)

(the index “0” means the contraction withyi).13 The canonical semisprayS is 2-homogeneous with respect toyi .

So,S is a spray.14 The nonlinear connectionN determined byS is also canonical

and it is exactly the famous Cartan nonlinear connection of thespaceFn. Its coefficients are

Nij(x,y) =

∂Gi(x,y)∂y j =

12

∂∂y j γ i

00(x,y). (2.1.10)

An equivalent form for the coefficientsNij is as follows

Nij = γ i

j0(x,y)−Cijk(x,y)γ

k00(x,y). (2.1.10’)

Consequently, we have

Ni0 = γ i

00 = 2Gi. (2.1.11)


Therefore, we can say: The semisprayS′ determined by the Cartannonlinear connectionN is the canonical spraySof spaceFn.

15 The Cartan nonlinear connectionN determines a splitting of vec-tor spaceTuTM, ∀u∈ TM of the form:

TuTM = Nu⊕Vu, ∀u∈ TM (2.1.12)

Thus, the adapted basis

(δ

δxi ,∂

∂yi

), (i = 1, ..,n), to the previous

splitting has the local vector fieldsδ

δxi given by:

δδxi =

∂∂xi −N j

i(x,y)∂

∂y j , i = 1, ..,n, (2.1.13)

with the coefficientsNij(x,y) from (2.1.6).

Its dual basis is(dxi ,δyi

), where

δyi = dyi +Nij(x,y)dxj . (2.1.14)

The autoparallel curves of the nonlinear connectionN are given by,[166],

d2xi

dt2+Ni

j(x,dxdt

)dxj

dt= 0. (2.1.15)

Using the dynamic derivative∇ defined byN, the equations (2.1.11)can be written as follows

∇(dxi

dt) = 0. (2.1.11’)

16 The variational equations of the autoparallel curves (2.1.11) givethe Jacobi equations:

∇2ξ i +

(∂Ni

j

∂yk

dxj

dt−Ni

k

)∇ξ k+Ri

jkdxj

dtdxk

dt= 0. (2.1.16)

The vector fieldξ i(t) along a solutionc(t) of the equations (2.1.11)and which verifies the previous equations is called a Jacobi field.

In the Riemannian case,∂gi j

∂yk = 0, the Jacobi equations (2.1.12) are

exactly the classical Jacobi equations:

2.1 Semidefinite Finsler spaces 175

∇2ξ i +Rijlk(x)

dxl

dtdxj

dtξ k = 0 (2.1.17)

17 A distinguished metric connectionsD with the coefficients

CΓ (N) =(

F ijk,C

ijk

)is defined as aN-linear connection onTM,

metric with respect to the fundamental tensorgi j (x,y) of FinslerspaceFn, i.e. we have

gi j |k =δgi j

δxk −Fsikgs j−Fs

jkgis = 0,

gi j |k =∂gi j

∂yk −Csikgs j−Cs

jkgis = 0.

(2.1.18)

18 The following theorem holds:

Theorem 2.1.1.1There is an unique N-linear connection D, withcoefficients CΓ (N) which satisfies the following system of ax-ioms: A1. N is the Cartan nonlinear connection of Finsler spaceFn.A2. D is metrical,(i.e. D satisfies(2.1.14)).A3. T i

jk = F ijk −F i

k j = 0, Sijk =Ci

jk −Cik j = 0.

2The metric N-linear connection D has the coefficients CΓ (N) =(F i

jk,Cijk

)given by the generalized Christoffel symbols

F ijk =

12

gis

(δgs j

δxk +δgsk

δx j −δg jk

δxs

),

Cijk =

12

gis

(∂gs j

∂yk +∂gsk

∂y j −∂g jk

∂ys

).

(2.1.19)

20 By means of this theorem, it is not difficult to see that we have

Cijk = gisCs jk (2.1.20)

andyi|k = 0. (2.1.21)

21 The Cartan nonlinear connectionN determines onTM an almostcomplex structureF, as follows:

F(δ

δxi ) =−∂

∂yi , F(∂

∂yi ) =δ

δxi , i = 1, ..,n. (2.1.22)


But one can see thatF is the tensor field onTM:

F=−∂

∂yi ⊗dxi +δ

δxi ⊗δyi , (2.1.22’)

with the 1-formsδyi and the vector fieldδ

δxi given by (2.1.10),

(2.1.9), (2.1.6).It is not difficult to prove that: The almost complex structure F isintegrable if and only if the distributionN is integrable onTM.

22 The Sasaki-Matsumoto lift of the fundamental tensorgi j of Fin-sler spaceFn is

G(x,y) = gi j (x,y)dxi ⊗dxj +gi j (x,y)δyi ⊗δy j . (2.1.23)

The tensor fieldG determines a pseudo-Riemannian structure onTM.

23 The following theorem is known:

Theorem 2.1.2.1 The pair(G,F) is an almost Hermitian struc-ture onTM determined only by the Finsler space Fn.2 The symplectic structure associate to the structure(G,F) is theCartan 2-form:

θ = 2gi j δyi ∧dxj . (2.1.24)

3 The space(TM,G,F) is almost Kahlerian.

The spaceH2n = (TM,G,F) is calledthe almost Kahlerian modelof the Finsler spaceFn.

G.S. Asanov in the paper [27] proved that the metricG from(2.1.23) does not satisfies the principle of the Post-Newtonian Cal-culus. This is due to the fact that the horizontal and vertical terms ofG do not have the same physical dimensions.

This is the reason for R. Miron to introduce a new lift of the funda-mental tensorgi j , [166], in the form:

G(x,y) = gi j (x,y)dxi ⊗dxj +a2

||y||2gi j (x,y)δyi ⊗δy j

where a > 0 is a constant imposed by applications in TheoreticalPhysics and where‖y‖2 = gi j (x,y)yiy j = F2 has the propertyF2 > 0.The liftG is 2-homogeneous with respect toyi . The Sasaki-Matsumotolift G has not the property of homogeneity.

2.2 The notion of Finslerian mechanical system 177

Two examples:1. Randers spaces.They have been defined by R. S.Ingarden as a tripleRFn = (M,α +β ,N), whereα +β is a Randersmetric andN is the Cartan nonlinear connection of the Finsler spaceFn = (M,α +β ), [175].

2. Ingarden spaces.These spaces have been defined by R. Miron,[166], as a tripleIF n = (M,α + β ,NL), whereα + β is a Randersmetric andNL is the Lorentz nonlinear connection ofFn = (M,α+β )having the coefficients

Nij(x,y)=

γ i

jk(x)yk−

F i

j(x),F i

j =12

ais(x)

(∂bs

∂x j −∂b j

∂xs

). (2.1.25)

The Christoffel symbols are constructed with the Riemannian met-

ric tensorai j (x) of the Riemann space(M,α2) andF i

j(x) is the elec-tromagnetic tensor determined by the electromagnetic form(α +β ).

2.2 The notion of Finslerian mechanical system

As we know from the previous chapter, the Riemannian mechanicalsystemsΣR = (M,T,Fe) is defined as a triple in whichM is the con-figuration space,T is the kinetic energy andFeare the external forces,which depend on the material pointx ∈ M and depend on velocities

yi =dxi

dt.

Extending the previous ideas, we introduce the notion of Finsle-rian Mechanical System, studied by author in the paper [175]. Theshortly theory of this analytical mechanics can be find in thejointbook Finsler-Lagrange Geometry. Applications to Dynamical Sys-tems, by Ioan Bucataru and Radu Miron, Romanian Academy Press,Bucharest, 2007.

In a different manner, M. de Leon and colab. [138], M. Crampinetcolab. [66], have studied such kind of new Mechanics.

A Finslerian mechanical systemΣF is defined as a triple

ΣF = (M,EF2,Fe) (2.2.1)

whereM is a real differentiable manifold of dimensionn, calledtheconfiguration space, EF2 is the energyof an a priori given FinslerspaceFn = (M,F(x,y)), which can be positive defined or semide-fined, andFe(x,y) are the external forces given as a vertical vector


field on the tangent manifoldTM. We continue to say thatTM is thevelocity space ofM.

Evidently, the Finslerian mechanical systemΣF is a straightfor-ward generalization of the known notion of Riemannian mechanicalsystemΣR obtained forEF2 as kinetic energy of a Riemann spaceRn = (M,g).

Therefore, we can introduce the evolution (or fundamental)equa-tions ofΣF by means of the following Postulate:

Postulate.The evolution equations of the Finslerian mechanical sys-temΣF are the Lagrange equations:

ddt

∂EF2

∂yi −∂EF2


dt(2.2.2)

where the energy is

EF2 = yi ∂F2

∂yi −F2 = F2, (2.2.3)

and Fi(x,y), (i = 1, ..,n), are the covariant components of the externalforces Fe:

Fe(x,y) = F i(x,y)∂

∂yi

Fi(x,y) = gi j (x,y)F i(x,y),

(2.2.4)

and

gi j (x,y) =12

∂ 2F2

∂yi∂y j , det(gi j (x,y)) 6= 0, (2.2.5)

is the fundamental (or metric) tensor of Finsler space Fn=(M,F(x,y)).Finally, the Lagrange equations of the Finslerian mechanical sys-

tem are:ddt

∂F2

∂yi −∂F2


dt. (2.2.6)

A more convenient form of the previous equations is given by:

Theorem 2.2.1.The Lagrange equations(2.2.6) are equivalent to thesecond order differential equations:

d2xi

dt2+ γ i

jk(x,dxdt

)dxj

dtdxk

dt=

12

F i(x,dxdt

), (2.2.7)

2.2 The notion of Finslerian mechanical system 179

whereγ ijk(x,y) are the Christoffel symbols of the metric tensor gi j (x,y)

of the Finsler space Fn.

Proof. Writing the kinetic energyF2(x,y) in the form:

F2(x,y) = gi j (x,y)yiy j , (2.2.8)

the equivalence of the systems of equations (2.2.6) and (2.2.7) is notdifficult to establish.

But, the form (2.2.7) is very convenient in applications. So, we ob-tain a first result expressed in the following theorems:

Theorem 2.2.2.The trajectories of the Finslerian mechanical systemΣF , without external forces(Fe≡ 0), are the geodesics of the Finslerspace Fn.

Indeed,F i(x,y) ≡ 0 and the SODE (2.2.7) imply the equations(2.2.4) of geodesics of spaceFn.

A second important result is a consequence of the Lagrange equa-tions, too.

Theorem 2.2.3.The variation of kinetic energyEF2 = F2 of the me-chanical systemΣF along the evolution curves (2.2.6) is given by

dEF2

dt=

dxi

dtFi . (2.2.9)

Consequently:

Theorem 2.2.4.The kinetic energyEF2 of the systemΣF is conservedalong the evolution curves (2.2.6) if the external forces Feare orthog-onal to the evolution curves.

The external forcesFe are calleddissipativeif the scalar product〈C,Fe〉 is negative, [175].

The formula (2.2.9) leads to the following property expressed by:

Theorem 2.2.5.The kinetic energyEF2 decreases along the evolutioncurves of the Finslerian mechanical systemΣF if and only if the exter-nal forces Fe are dissipative.


Some examples of Finslerian mechanical systems

1 The systemsΣF = (M,EF2,Fe) given byFn = (M,α+β ) as a Ran-

ders space andFe= βC = βyi ∂∂yi . EvidentlyFe is 2-homogeneous

with respect toyi .2 ΣF determined byFn = (M,α +β ) andFe= αC.3 ΣF with Fn = (M,α +β ) andFe= (α +β )C.

4 ΣF defined by a Finsler spaceFn=(M,F) andFe=aijk(x)y

jyk ∂∂yi ,

aijk(x) being a symmetric tensor on the configuration spaceM of type

(1,2).

2.3 The evolution semispray of the systemΣF

The Lagrange equations (2.2.6) give us the integral curves of a re-markable semispray on the velocity spaceTM, which governed thegeometry of Finslerian mechanical systemΣF . So, if the externalforcesFeare global defined on the manifoldTM, we obtain:

Theorem 2.3.1 (Miron, [175]). For the Finslerian mechanical sys-temsΣF , the following properties hold good:

1 The operator S defined by

S= yi ∂∂xi −

(2

G i −

12

F i)

∂∂yi ; 2

G i = γ i

jk(x,y)yjyk (2.3.1)

is a vector field, global defined on the phase space TM.2 S is a semispray which depends only onΣF and it is a spray if

Fe is2-homogeneous with respect to yi .3 The integral curves of the vector field S are the evolution curves

given by the Lagrange equations (2.2.7) ofΣF .

Proof. 1 Let us consider the canonical semisprayS of the Finsler

spaceFn. Thus from (2.3.1) we have

S=S+

12

Fe. (2.3.2)

It follows thatS is a vector field onTM.

2.4 The canonical nonlinear connection of the Finslerian mechanical systemsΣF 181

2 SinceFe is a vertical vector field, thenS is a semispray. Evi-dently,Sdepends onΣF , only.

3 The integral curves ofSare given by:

dxi

dt= yi ;

dyi

dt+2

Gi (x,y) =12

F i(x,y). (2.3.3)

The previous system of differential equations is equivalent to sys-tem (2.2.7).

In the book of I. Bucataru and R. Miron [49], one proves the fol-lowing important result, which extend a known J. Klein theorem:

Theorem 2.3.2.The semispray S, given by the formula(2.3.1), is theunique vector field onTM, solution of the equation:

iSω=−dT+σ , (2.3.4)

whereω is the symplectic structure of the Finsler space Fn = (M,F),

T =12

F2 =12

gi jdxi

dtdxj

dtandσ is the 1-form of external forces:

σ = Fi(x,y)dxi . (2.3.5)

In the terminology of J. Klein, [123],S is the dynamical system ofΣF , defined on the tangent manifoldTM. We will say thatS is theevolution semispray ofΣF .

By means of semisprayS(or sprayS) we can develop the geometryof the Finslerian mechanical systemΣF . So, all geometrical notionderived fromS, as nonlinear connections, N-linear connections etc.will be considered as belong to the systemΣF .

2.4 The canonical nonlinear connection of theFinslerian mechanical systemsΣF

The evolution semisprayS(2.3.1) has the coefficientsGi expressed by

2Gi(x,y) = 2

Gi (x,y)−12

F i(x,y) = γ ijk(x,y)y

jyk−12

F i(x,y).

(2.4.1)Thus, the evolution nonlinear connectionN (or canonical nonlinear

connection) of the Finslerian mechanical systemΣF has the coeffi-


cients:

Nij =

∂Gi

∂y j =

Nij −

14

∂F i

∂y j (2.4.2)

whereN with coefficients

Nij is the Cartan nonlinear connection of

Finsler spaceFn = (M,F(x,y)).N depends on the mechanical systemΣF , only. It is called canonical

for Fn.The nonlinear connectionN determines the horizontal distribution,

denoted byN too, with the property

TuTM = Nu⊕Vu, ∀u∈ TM, (2.4.3)

Vu being the natural vertical distribution on the tangent manifold TM.A local adapted basis to the horizontal and vertical vector spaces

Nu andVu is given by

(δ

δxi ,∂

∂yi

), (i = 1, ..,n), where

δδxi =

∂∂xi −N j

i∂

∂y j =∂

∂xi −

(N j

i −14

∂F j

∂yi

)∂

∂y j , i = 1, ..,n,

(2.4.4)and the adapted cobasis(dxi ,δyi) with

δyi = dyi +Nijdxj = dyi +

(Ni

j −14

∂F i

∂y j

)dxj , i = 1, ..,n. (2.4.4’)

From (2.4.4) and (2.4.4’) it follows

δδxi =

δδxi +

14

∂F j

∂yi

∂∂y j ,

δyi =

δyi −14

∂F i

∂y j dxj .

(2.4.5)

Now we easy determine the curvatureR ijk and torsiont i

jk of thecanonical nonlinear connectionN. One obtains

2.4 The canonical nonlinear connection of the Finslerian mechanical systemsΣF 183

R ijk =

δNij

δxk −δNi

k

δx j =

(δ

δxk

∂∂y j −

δδx j

∂∂yk

)(

Gi −12F i),

t ijk =

∂Nij

∂yk −∂Ni

k

∂y j = 0

(2.4.6)

such that:1 The torsion of the canonical nonlinear connectionN vanishes.2 The conditionR i

jk = 0 is necessary and sufficient forN to beintegrable.

Another important geometric object field determined by the canon-ical nonlinear connectionN is the Berwald connectionBΓ (N) =

(Bijk,0) =

(∂Ni

j

∂yk ,0

). Its coefficients are:

Bijk =

Bi

jk −14

∂ 2F i

∂y j∂yk , (2.4.7)

whereBi

jk are coefficients of Berwald connection of Finsler spaceFn.The coefficients (2.4.7) are symmetric.

We can prove:Theorem 2.4.1.The autoparallel curves of the evolution nonlinearconnection N are given by the following SODE:

yi =dxi

dt,

δyi

dt=

δ yi

dt−

14

∂F i

∂y j

dxj

dt= 0. (2.4.8)

Corollary 2.4.1. If the external forces Fe vanish, then the evolutionnonlinear connection N is the Cartan nonlinear connection of Finslerspace Fn.

Corollary 2.4.2. If the external forces Fe are2-homogeneous with re-spect to velocities yi , then the equations(2.4.8) coincide with the evo-lution equations of the Finslerian mechanical systemΣF .

It is not difficult to determine the evolution nonlinear connectionNfrom examples in section 2.3.

Lemma 2.4.1.The exterior differential of the1-formsδyi are givenby formula:

d(δyi) = dNij ∧dxj =

12R

ik jdxj ∧dxk+Bi

k jδy j ∧dxk. (2.4.9)


2.5 The dynamical derivative determined by theevolution nonlinear connectionN

The dynamical covariant derivation induced by the evolution nonlin-ear connectionN is expressed by (?) using the coefficientsNi

j from(2.4.2). It is given by

∇(

Xi ∂∂yi

)=(SXi +X jNi

j) ∂

∂yi . (2.5.1)

Applied to ad-vector fieldXi(x,y), we have the formula:

∇Xi = SXi| = SXi +X jNi

j (2.5.2)

and for ad-tensorgi j , we have

∇gi j = gi j | = Sgi j −gs jNsi −gisN

sj , (2.5.2’)

where

S=S+

12

Fe

Nij =

N i

j −14

∂F i

∂y j .

(2.5.3)

Remarking that∂F i

∂y j is ad-tensor field of type (1,1), one can intro-

duce twod-tensors, important in the geometrical theory of the Finsle-rian mechanical systemsΣF :

Pi j =12

(∂Fi

∂y j −∂Fj

∂yi

), Qi j =

12

(∂Fi

∂y j +∂Fj

∂yi

). (2.5.4)

The first onePi j is called the helicoidal tensor of the Finslerianmechanical systemΣF , [166].

Also, on the phase spaceTM, the elicoidald-tensorPi j give rise tothe 2-form

P= Pi j dxi ∧dxj (2.5.5)

andQi j allows to consider the symmetric tensor

Q= Qi j dxi ⊗dxj . (2.5.6)

2.6 Metric N-linear connection ofΣF 185

The following Bucataru-Miron theorem holds:

Theorem 2.5.1.For a Finslerian mechanical systemΣF =(M,EF2,Fe)the evolution nonlinear connection N is the unique nonlinear connec-tion that satisfies the conditions:

∇g=−12

Q,ω (hX,hY) =

12

P(X,Y), ∀X,Y ∈ χ(TM), (2.5.7)

whereω is the symplectic structure of the Finsler space Fn:

ω= 2gi j

δ y j ∧dxi . (2.5.8)

If Fe does not depend on velocitiesyi =dxi

dtwe haveP ≡ 0, Q≡

0. One obtains the the case of Riemannian mechanical systemsΣR,studied in the previous chapter.

2.6 Metric N-linear connection ofΣF

The metric, or canonic,N-linear connectionD, with coefficientsCΓ (N)= (F i

jk,Cijk) of the Finslerian mechanical systemΣF is uniquely

determined by the following axioms, (Miron [161]):A1. N is the canonical nonlinear connection ofΣF .A2. D is h-metric, i.egi j |k = 0.A3. D is h-symmetric, i.e.T i

jk = F ijk −F i

k j = 0.A4. D is v-metric, i.e.gi j |k = 0.A5. D is v-symmetric, i.e.Si

jk =Cijk −Ci

k j = 0.The following important result holds:

Theorem 2.6.1.The local coefficients DΓ (N) = (F ijk,Ci

jk) of thecanonical N-connection D of the Finslerian mechanical system ΣFare given by the generalized Christoffel symbols:

F ijk =

12

gis

(δgsk

δx j +δg js

δxk −δg jk

δxs

),

Cijk =

12

gis

(∂gsk

∂y j +∂g js

∂yk −∂g jk

∂ys

).

(2.6.1)


Using the expression (?) of the operatorδ

δxi , one obtains:

Theorem 2.6.2.The coefficients Fi jk, Cijk of canonical N-connection

D have the expressions:

F ijk =

F i

jk +Cijk, Ci

jk =C i

jk

Cijk =

14

gis

(Cskh

∂Fh

∂y j +C jsh

∂Fh

∂yk −C jkh

∂Fh

∂ys

),

(2.6.2)

where CΓ (N) = (

F i

jk,C i

jk) are the coefficients of canonic Cartanmetric connection of Finsler space Fn.

A consequence of the previous formulas is the following relation:

y jCijh =

14

C i

hkys∂Fk

∂ys . (2.6.3)

Let ω ij be the connection forms ofCΓ (N):

ω ij = F i

jkdxk+Cijkδyk. (2.6.4)

Then, we have

Theorem 2.6.3.The structure equations of the canonical connectionCΓ (N) are given by:


1Ω i ,


2Ω i ,

dω ij − ωk

j ∧ω ik =− Ω i

j ,

(2.6.5)

where the2-forms of torsion1

Ω i ,2

Ω i are as follows

1Ω i =Ci

jkdxj ∧δyk,

2Ω i =

12

Rijkdxj ∧dxk+Pi

jkdxj ∧δyk,

(2.6.6)

with

2.6 Metric N-linear connection ofΣF 187

Rijk =

δNij

δxk −δNi

k

δx j , Pijk =

∂Nij

∂yk −F ik j (2.6.7)

and the2-form of curvatureΩ ij is given by

Ω ij =

12

R ij kh dxk∧dxh+P i

j kh dxk∧δyh+12

S ij kh δyk∧δyh, (2.6.8)

where

R ij kh =

δF ijk

δxh −δF i

jh

δxk +FsjkF i

sh−FsjhF i

sk+CihsRs

kh,

P ij kh =

∂F ijk

∂yh −Cijh |k+Ci

jsPskh,

S ij kh =

∂Cijk

∂yh −∂Ci

jh

∂yk +CsjkC

ish−Cs

jhCijk.

(2.6.9)

Taking into account that the coefficientsF ijk andCi

jk are expressedin the formulas (2.6.2), the calculus of curvature tensors is not diffi-cult.

Exterior differentiating (2.6.5) and using them again one obtainsthe Bianchi identities ofCΓ (N).

The h- and v-covariant derivatives, denoted by “|” and “|”, withrespect to canonical connectionD have the properties given by theaxiomsA1−A4.

So, we obtain

gi j |k =δgi j

δxk −gs jFsik −gisFs

jk = 0,

gi j |k =∂gi j

∂yk −gs jCsik −gisCs

jk = 0.

(2.6.10)

The Ricci identities applied to the fundamental tensorgi j give us:

Ri jkh+Rjikh = 0, Pi jkh+Pjikh = 0, Si jkh+Sjikh = 0,

whereRi jkh = g jsRiskh, etc.

Also, Pi jk = gisPsjk is totally symmetric.

The deflection tensors ofD are


Dij = yi

| j =δyi

δx j +ysF is j; di

j = yi | j . (2.6.11)

Taking into account (2.6.11), we obtain:

Dij = ysCi

s j+14

∂F i

∂y j , dij = δ i

j . (2.6.12)

2.7 The electromagnetism in the theory of theFinslerian mechanical systemsΣF

For a Finslerian mechanical systemΣF = (M,EF2,Fe) whose externalforcesFe depend on the material pointx ∈ M and on velocityyi =dxi

dt, the electromagnetic phenomena appears because the deflection

tensorsDij anddi

j nonvanish.SettingDi j = gihDh

j , di j = gihδ hj , theh-electromagnetic tensorFi j

and thev-electromagnetic tensorfi j are defined by


12(di j −d ji ). (2.7.1)

By using the equalities (2.6.11), we have

Fi j =14

Pi j , fi j = 0. (2.7.2)

If we denoteRi jk := gihRhjk, then one proves:

Theorem 2.7.1.The electromagnetic tensorFi j of the Finslerian me-chanical systemΣF = (M,T,Fe) satisfies the following generalizedMaxwell equations:

2.8 The almost Hermitian model on the tangent manifoldTM of the Finslerian mechanical systemsΣF189

Fi j |k+F jk |i +Fki | j =12ys(Rsi jk+Rs jki+Rski j)−

−(Ri jk +Rjki +Rki j),

Fi j |k+F jk|i +Fki| j =12ys[(Psi jk−Psik j)+

+(Ps jki−Ps jik)+(Pski j−Psk ji)]

(2.7.3)

whereFi j is expressed in (2.7.2).

Corollary 2.7.3. If the external forces Fe does not depend on the ve-locity yi then the electromagnetic fieldsFi j and fi j vanish.

Remark 2.7.1.1 The application of the previous theory to the exam-ples from section 2.2 is immediate.

2 The theory of the gravitational fieldgi j and the Einstein equa-tions can be realized by same method as in the papers, [166].

2.8 The almost Hermitian model on the tangentmanifold TM of the Finslerian mechanical systemsΣF

Consider a Finslerian mechanical systemΣF = (M,F(x,y),Fe(x,y))endowed with the evolution nonlinear connectionN and also endowedwith the canonicalN−metrical connection having the coefficientsCΓ (N) = (F i

jk,Cijk). On the velocity manifoldTM = TM\ 0 we

can see that the previous geometrical object fields determine an al-most Hermitian structureH2n = (TM,G,F). Moreover, the theory ofgravitational and electromagnetic fields can be geometrically studiedmuch better on such model, since the symplectic structureθ , the al-most complex structureF and the Riemannian structureG on TM arewell determined by the Finslerian mechanical systemΣF .

One obtains:


G= gi j dxi ⊗dxj +gi j δyi ⊗δy j

F=−∂

∂yi ⊗dxi +δ

δxi ⊗δyi

P=−∂

∂yi ⊗δyi +δ

δxi ⊗dxi ,

(2.8.1)

where

δδxi =

δδxi +

14

∂Fs

∂yi

∂∂ys

δyi =

δ yi −14

∂F i

∂ys dxs.

(2.8.2)

Thus, the following theorem holds:

Theorem 2.8.1.We have:1 The pair(TM,G) is a pseudo Riemannian space.2 The tensorG depends onΣF , only.3 The distributions N and V are orthogonal with respect toG.

Theorem 2.8.2.1 The pair(TM,F) is an almost complex space.2 F depends onΣR, only.3 F is integrable onTM iff the tensors Ri jk vanishes.

Theorem 2.8.3.1 The pair(TM,P) is an almost product space.2 P depends onΣF , only.3 P is integrable iff the tensors Ri jk vanishes.

The integrability ofF andP are studied by means of Nijenhuis ten-sorsNP andNF :

NF(X,Y) = F2(X,Y)+ [FX,FY]−F [FX,Y]−

−F [X,FY] , ∀X,Y ∈ χ(TM).

In the adapted basis

(δ

δxi ,∂

∂yi

)NF=0 if, and only ifRi

jk =0, t ijk =

0. But the torsion tensort ijk vanishes.

It is not difficult to prove the following results:

Theorem 2.8.4.1 The triple(

TM,G,F)

is an almost Hermitianspace.

2.8 The almost Hermitian model on the tangent manifoldTM of the Finslerian mechanical systemsΣF191

2(

TM,G,F)

depends on Finslerian mechanical systemΣF , only.

3 The almost symplectic structureθ of the space(

TM,G,F)

is

θ = gi j δyi ∧dxj , (2.8.3)

with δyi from (2.8.2).

If the almost symplectic structureθ is a symplectic one (i.e.dθ =

0), then the spaceH2n =(

TM,G,F)

is almost Kahlerian.

But, using the formulas (2.8.3) and (2.4.9) one obtains:

dθ =13!(Ri jk +Rjki +Rki j)dxi ∧dxj ∧dxk+

+12(gisBs

jk −g jsBsik)δyk∧dxj ∧dxi .

(2.8.4)

Therefore we deduce

Theorem 2.8.5.The space H2n =(

TM,G,F)

is almost Kahlerian if,

and only if the following equations hold:

Ri jk +Rjki +Rki j = 0; gisBs

jk −g jsBsik = 0. (2.8.5)

The spaceH2n =(

TM,G,F)

is called the almost Hermitian model

of the Finslerian Mechanical SystemΣF .

Remark. We can study the spaceH2n =(

TM,G,P)

by similar way.

One can use the modelH2n =(

TM,G,F)

to study the geometrical

theory of Finslerian Mechanical systemΣF = (M,F(x,y),Fe(x,y)).For instance, the Einstein equations of the pseudo Riemannian space(TM,G) can be considered as the Einstein equations of the FinslerianMechanical systemΣF .

Remark.G.S. Asanov showed, [27], that the metricG given by the lift(2.8.1) does not satisfy the principle of the Post-Newtonian calculus.This is due to the fact that the horizontal and vertical termsof themetricG do not have the same physical dimensions. This is the reasonfor the author to introduce a new lift, [174], of the fundamental tensorgi j (x,y) of ΣF that can be used in a gauge theory. This lift is similarwith that introduced in section 2.1, adapted toΣR.

It is expressed by


G(x,y) = gi j (x,y)dxi ⊗dxj +a2

F2gi j (x,y)δyi ⊗δy j , (2.8.6)

whereδyi =

δ yi +14

∂F i

∂y j dxj anda> 0 is a constant imposed by appli-

cations. This is to preserve the physical dimensions to the both termsof G.

Let us consider also the tensor field onTM

F =−Fa

∂∂yi ⊗dxi +

aF

δδxi ⊗δyi (2.8.7)

and the 2-formθ =

aF

θ . (2.8.8)

Then, we have:

Theorem 2.8.6.1 The triple

(TM,G, F

)is an almost Hermitian space.

2 The2-form θ given by (2.8.8) is the almost symplectic structure

determined by(G, F

).

3 θ is conformal toθ .

The spaceH2n =(

TM,G, F)

can be used to study the geometrical

theory of the Finslerian mechanical systemΣF , too.

Chapter 3Lagrangian Mechanical systems

A natural extension of the notion of the Finslerian mechanical systemis that of the Lagrangian mechanical system. It is defined as atripleΣL = (M,L(x,y),Fe(x,y)) where:M is a realn−dimensionalC∞ man-ifold called the configuration space;L(x,y) is a regular Lagrangianwith the property that the pairLn = (M,L(x,y)) is a Lagrange space;Fe(x,y) is an a priori given vertical vector field on the velocity spaceTM called the external forces. The numbern is the number of freedomdegree ofΣL. The equations of evolution, or fundamental equations ofLagrangian mechanical systemΣL are the Lagrange equations:

(I)ddt

∂L∂yi −


dxi

dt(i = 1,2, ...,n)

whereFi are the covariant components of the external forcesFe. Theseequations determine the integral curves of a canonical semisprayS.Thus, the geometrical theory of the semispraySis the geometrical the-ory of the Lagrangian mechanical systemΣL. The LagrangianL(x,y)and the external forcesFe(x,y) do not explicitly depend on the timet.ThereforeΣL is a scleronomic Lagrangian mechanical system.

3.1 Lagrange Spaces. Preliminaries

Let Ln = (M,L(x,y)) be a Lagrange space (see ch. 2, part I).L : TM→R being a regular Lagrangian for which

gi j (x,y) =12

∂i ∂ jL(x,y) (3.1.1)

193

194 3 Lagrangian Mechanical systems

is the fundamental tensor. Thusgi j (x,y) is a covariant of order 2 sym-metricd−tensor field of constant signature and non singular:

det(gi j (x,y)) 6= 0 onTM = TM\0. (3.1.2)

The Euler-Lagrange equations of the spaceLn are:

ddt

∂L∂yi −

∂L∂xi = 0, yi =

dxi

dt. (3.1.3)

As we know, the system of differential equations (3.1.3) canbewritten in the equivalent form:

d2xi

dt2+2

Gi(

x,dxdt

)= 0 (3.1.4)

with

2G

i=

12

gis(∂s∂hL)yh−∂sL (3.1.4’)(

∂i =∂

∂yi ,∂i =∂

∂xi

).

G

iare the coefficients of a semispray

S:

S= yi∂i −2

G

i(x,y)∂i (3.1.5)

andSdepend on the spaceLn, only.

S is called the canonical semispray of Lagrange spaceLn.The energy of the LagrangianL(x,y) is:

EL = yi ∂iL−L. (3.1.6)

In the chapter 2, part I, we prove the Law of conservation energy:

Theorem 3.1.1.Along the solution curves of Euler-Lagrange equa-tions(3.1.3) the energyEL is conserved.

The following properties hold:

1The canonical nonlinear connectionN of Lagrange spaceLn has the

coefficients

3.1 Lagrange Spaces. Preliminaries 195

N

i

j = ∂ jG

i(3.1.7)

2N determine a differentiable distribution on the velocity spaceTMsupplementary to the vertical distributionV:

TuTM =Nu⊕Vu, ∀u∈ TM. (3.1.8)

3An adapted basis to (3.1.8) is(

δ i , ∂i)u, (i = 1, ...,n), where

δ i = ∂i −N

j

i (x,y)∂ j (3.1.9)

and an adapted cobasis(dxi ,

δyi)u, (i = 1, ...,n) with

δyi = dyi +Nij(x,y)dxj (3.1.9’)

4The canonical metricalN−connectionCΓ (

N) = (

L

i

jk,C

i

jk) has thecoefficients expressed by the generalized Christoffel symbols:

L

i

jh =12

gis(

δ jgsh+

δ hg js−

δ sg jh)

C

i

jh =12

gis(∂ jgsh+ ∂hg js− ∂sg jh)

(3.1.10)

5The Cartan 1-form ofLn is

ω =

12(∂iL)dxi (3.1.11)

6The Cartan-Poincare 2-form ofLn is

θ = d

ω = gi j (x,y)

δyi ∧dxj (3.1.12)

7The 2-formθ determine a symplectic structure on the velocity man-

ifold TM.

Example 3.1.1.The function



2em

Ai(x)yi +U (x) (3.1.13)

with m,c,e the known physical constants,γ i j (x) are the gravitationalpotentials,γ i j (x) being a Riemannian metric,Ai(x) are the electro-magnetic potentials andU (x) is a potential function. Thus,Ln =(M,L(x,y)) is a Lagrange space. It is the Lagrange space of electro-dynamics (cd. ch. 2, part I).

3.2 Lagrangian Mechanical systems,ΣL

Definition 3.2.1.A Lagrangian mechanical system is a triple:

ΣL = (M,L(x,y),Fe(x,y)), (3.2.1)

whereLn = (M,L(x,y)) is a Lagrange space,Fe(x,y) is an a priorigiven vertical vector field:

Fe(x,y) = F i(x,y)∂i , (3.2.2)

the numbern = dimM is the number of freedom degree ofΣL; themanifoldM is realC∞ differentiable manifold called the configurationspace;Fe(x,y) is the external forces andF i(x,y) are the contravariantcomponents ofFe. Of course,F i(x,y) is a vertical vector field and

Fi(x,y) = gi j (x,y)Fj(x,y) (3.2.3)

are the covariant components ofFe. Fi(x,y) is ad−covector field.The 1-form:

σ = Fi(x,y)dxi (3.2.4)

is a vertical 1-form on the velocity spaceTM.

The Riemannian mechanical systemsΣR and the Finslerian me-chanical systemsΣF are the particular Lagrangian mechanical sys-temsΣL.

Remark 3.2.1.The notion of Lagrangian mechanical systemΣL is dif-ferent of that of “Lagrangian mechanical system” defined by JosephKlein [123], which is a Riemannian conservative mechanicalsystem.

The fundamental equations ofΣL are an extension of the Lagrangeequations of a Finslerian mechanical systemΣF , from the previouschapter.

3.3 The evolution semispray ofΣL 197

So, we introduce the following Postulate:

Postulate.The evolution equations of the Lagrangian mechanical sys-temΣL = (M,L,Fe) are the following Lagrange equations:

ddt

(∂L∂yi

)−


dxi

dt. (3.2.5)

But the both members of the Lagrange equations (3.2.5) ared−co-vectors. Consequently, we have:

Theorem 3.2.1.The Lagrange equations(3.2.5) of a Lagrangian me-chanical systemΣL = (M,L,Fe) have a geometrical meaning.

Theorem 3.2.2.The trajectories without external forces of the La-grangian mechanical systemΣL = (M,L,Fe) are the geodesics of theLagrange space Ln = (M,L).

Indeed,Fi(x,y)≡ 0 implies the previous affirmations.For a regular LagrangianL(x,y) (in this case (3.1.2) holds), the La-

grange equations (3.2.5) are equivalent to the second orderdifferentialequations (SODE):

d2xi

dt2+2

G

i(x,

dxdt

)=

12

F i(

x,dxdt

), (3.2.6)

where the functionsG

i(x,y) are the local coefficients (3.1.4’) of

canonical semispraySof Lagrange spaceLn = (M,L(x,y)).

The equations (3.2.6) are called fundamental equations ofΣL, too.

3.3 The evolution semispray ofΣL

The Lagrange equations (3.2.6) determine a semisprayS which de-pend on the Lagrangian mechanical systemΣL, only.

Indeed, the vector field onTM:

S= yi∂i −2Gi(x,y)∂i (3.3.1)

with

2Gi(x,y) = 2G

i(x,y)−

12

F i(x,y) (3.3.2)


has the propertyJS= C. So it is a semispray depending only onΣL.

Theorem 3.3.1 (Miron).For a Lagrangian mechanical systemΣL thefollowing properties hold:

1The semispray S is given by

S=S+

12

Fe (3.3.1’)

2S is a dynamical system on the velocity space TM.3The integral curves of S are the evolution curves(3.2.6) of ΣL.

In fact, 1 derives from the formulas (3.3.1) and (3.3.2).2 Sbeing a vector field onTM, compatible with the geometric struc-ture of TM, (i.e. JS= C), it is a dynamical system on the manifoldTM.3 The integral curves ofSare determined by the system of differen-tial equations

dxi

dt= yi ,

dyi

dt+2Gi(x,y) = 0. (3.3.3)

By means of the expression (3.3.2) of the coefficientsGi(x,y) the sys-tem (3.3.4) is coincident to (3.2.6).

The vector fieldS is called theevolution(or canonical) semisprayof the Lagrangian mechanical systemΣL. Exactly as in the case ofRiemannian or Finslerian mechanical systems, one can prove:

Theorem 3.3.2 ([49]).The evolution semispray S is the unique vectorfield, on the velocity space TM, solution of the equation

iSθ =−dEL +σ (3.3.4)

with L from (3.1.6) andσ from (3.2.4).

But Sbeing a solution of the previous equations, we get:

S(ΣL) = dEL(S) = σ(S) = Fiyi = gi j F

iy j .

So, we have:

Theorem 3.3.3.The variation of energyEL along the evolution curvesof mechanical systemΣL is given by

dEL

dt= Fi

(x,

dxdt

)dxi

dt. (3.3.5)

3.4 The evolution nonlinear connection ofΣL 199

The external forces fieldFe is called dissipative ifg(C,Fe) =gi j F iy j ≤ 0. Thus, the previous theorem implies:

Theorem 3.3.4.The energy of Lagrange space Ln = (M,L) is de-creasing along the evolution curves of the mechanical system ΣL ifand only if the external forces field Fe is dissipative.

Evidently, the semispraySbeing a dynamical system on the veloc-ity spaceTM it can be used for study the important problems, as thestability of evolution curves ofΣL, the equilibrium points etc.

3.4 The evolution nonlinear connection ofΣL

The geometrical theory of the Lagrangian mechanical systemΣL isbased on the evolution semisprayS, with the coefficientsGi(x,y) ex-pressed in formula (3.3.2). Therefore, all notions or properties derivedfrom S will be considered belonging to the mechanical systemΣL.Such that, theevolution nonlinear connection Nof ΣL is character-ized by the coefficients:

Nij(x,y)= ∂ jG

i(x,y)= ∂G(x,y)−

14

∂F i(x,y)=N

i

j(x,y)−14

∂ jFi(x,y).

(3.4.1)Since∂ jFi is ad−tensor field, consider its symmetric and skewsym-

metric parts:

Pi j =12(∂ jFi + ∂iFj), Fi j =

12(∂ jFi − ∂iFj) (3.4.2)

Thed−tensorFi j is theelicoidal tensorfield of ΣL, [166].The evolution nonlinear connectionN allows to determine the dy-

namic derivative of the fundamental tensorgi j (x,y) of ΣL:

gi j | = S(gi j )−gi|sNsj −gs jN

si . (3.4.3)

It is not difficult to prove the following formula:

gi j | =12

Pi j . (3.4.4)

N is metric nonlinear connection ifgi j | = 0. So, we have


Proposition 3.4.1.The evolution nonlinear connection N is metric if,and only if the d−tensor field Pi j vanishes.

The adapted basis(δ i , ∂i) to the distributionsN andV has the oper-atorsδ i of the form:

δ i = ∂i −N ji ∂ j =

δ i +14

∂iFs∂s. (3.4.5)

The dual adapted cobasis(dxi ,δyi) has 1-formsδyi :

δyi = dyi +Nijdxj =

δyi −14

∂ jFidxj (3.4.5’)

In the adapted basis the evolution semisprayShas the expression:

S= yi δδxi −

[2G

i−

N

i

jyj −

12

(F i −

12

∂ jFiy j)]

∂∂yi . (3.4.5”)

Consequences:

1Scannot be a vertical vector field.

2 If the coefficientsG

iare 2-homogeneous with respect to the ver-

tical variablesyi and the contravariant componentsF i(x,y) are 2-homogeneous inyi , then the canonical semisprayS belongs to thehorizontal distributionN.

The tensor of integrability of the distributionN is

Rijh = δ hNi

j −δ jNih = (δ h∂ j −δ j ∂h)

(G

i−

14

F i)

(3.4.6)

Thus:N is integrable iffRijh = 0.

Thed−tensor of torsion of the nonlinear connectionN vanishes.Indeed:

t ijh = ∂hNi

j − ∂ jNih = 0. (3.4.7)

The Berwald connectionBΓ (N) = (Bijk,0) of the canonical nonlin-

ear connectionN has the coefficients

Bijh =

B

i

jh −14

∂ j ∂hF i , (3.4.8)

3.5 CanonicalN−metrical connection ofΣL. Structure equations 201

where(B

i

jh,0) are the coefficients of Berwald connection of LagrangespaceLn.

It follows thatBΓ (N) is symmetric:Bijh−Bi

h j = t ijh = 0.

In the following section we need:

Lemma 3.4.1.The exterior differential of 1-formsδyi are given by

dδyi =12

Rijhdxh∧dxj +Bi

jhδy j ∧dxh. (3.4.9)

Indeed, from (3.4.5’) we have:

dδyi = dNij ∧dxj = δ hNi

jdxh∧dxj + ∂hNijδyh∧dxj ,

which are exactly (3.4.9).By means of the formula (3.4.5’) one gets:

Theorem 3.4.1.The autoparallel curves of the canonical nonlinearconnection N are given by the differential system of equations:

yi =dxi

dt,

δyi

dt=

δyi

dt−

14

∂ jFi dxj

dt= 0. (3.4.10)

Evidently, if Fe= 0, then the canonical nonlinear connectionN

coincides with the canonical nonlinear connectionN of the Lagrange

spaceLn.In particular, ifΣL is a Finslerian mechanical systemΣF , then the

previous theory reduces to that studied in the previous chapter.

3.5 CanonicalN−metrical connection ofΣL. Structureequations

Taking into account the results from part I, the canonicalN−metricalconnectionCΓ (N) is characterized by:

Theorem 3.5.1.The local coefficients DΓ (N)= (Lijk,C

ijk) of the canon-

ical N−metrical connection of Lagrangian mechanical systemΣL aregiven by the generalized Christoffel symbols:


Lijh =

12

gis(δ jgsh+δ hgs j−δ sg jk)

Cijh =

12

gis(∂ jgsh+ ∂hgs j− ∂sg jh).

(3.5.1)

Using the expression (3.4.5) of the operatorδ k and developing thetermsδ hgi j from (3.5.1), we get:

Theorem 3.5.2.The coefficients Lijh,Cijh of DΓ (N) can be set in the

form

Lijk =

L

i

jk +Cijk, Ci

jk =C

i

jk,

Cijk =

14

gis

(Cskh∂Fh+

C jsh∂Fh−

C jkh∂Fh

),

(3.5.2)

where CΓ (N) = (

L

i

jk,C

i

jk) is the canonical N−metrical connection ofthe Lagrange space Ln = (M,L(x,y)).

Let ω ij be the connection 1-forms of the canonicalN−metrical con-

nectionDΓ (N):ω i

j = Lijkdxk+Ci

jkδyk. (3.5.3)

Then, we have:

Theorem 3.5.3.The structure equations of canonical N−metrical con-nection DΓ (N) of Lagrangian mechanical systemΣL are given by:


1Ω

i


2Ω

i

dω ij −ωk

j ∧ω ik =−Ω i ,

(3.5.4)

where the 1-form of torsions1

Ωi

and2

Ωi

are:

3.5 CanonicalN−metrical connection ofΣL. Structure equations 203

1Ω

i

=Cijkdxj ∧δyk

2Ω

i

=12

Rijkdxj ∧dxk+Pi

jkdxj ∧δyk

(3.5.5)

and the 2-forms of curvatureΩ ij is as follows:

Ω ij =

12

Rijkhdxk∧dxh+P i

j khdxk∧δyh+12

Sij khδyk∧δyh (3.5.6)

where the d−tensors of torsions are:

T ijk = Li

jk −Lik j = 0; Si

jk =Cijk −Ci

k j = 0,

Cijk =

C

i

jk, Rijk = δ kNi

j −δ jNik, Pi

jk = Bijk −Li

jk

(3.5.7)

and the d−tensors of curvature are:

R ij kh = δ kLi

jk −δ kLijh +Ls

jkLish−Ls

jhLisk+Ci

jsRskh

P ij kh = ∂hLi

jk −Cijh|k+Ci

jsPskh

S ij kh = ∂hCi

jk − ∂kCijh+Cs

jkCish−Cs

jhCisk.

(3.5.8)

Here Cijk|h is the h−covariant derivative of the tensor Cijk with respect

to DΓ (N).

So, we have

gi j |h = δ hgi j −gs jLsih−gisLs

jh = 0,

gi j |h = ∂hgi j −gs jCsih −gisCs

jh = 0.(3.5.9)

Applying the Ricci identities to the fundamental tensorgi j , it is notdifficult to prove the identities

Ri jkh+Rjikh = 0, Pi jkh+Pjikh = 0, Si jkh +Sjikh = 0, (3.5.10)

whereRi jkh = g jsR si kh, etc.

Also, Pi jk = gisPskh is totally symmetric.


Taking into account the form (3.5.2) of the coefficients ofDΓ (N),the calculus ofd−tensors of torsion (3.5.7) and ofd−tensors of cur-vature (3.5.6) can be obtained.

The exterior differentiating the structure equations (3.5.4) modulothe same system of equations (3.5.4) and using Lemma (3.4.1)oneobtains the Bianchi identities of theN−metrical connectionDΓ (N).

The h− andv−covariant derivative of Liouville vector fieldC =

yi ∂∂yi lead to the deflection tensors ofDΓ (N):

Dij = yi

| j = δ jyi +ysLi

s j, dij = yi | j = ∂ jy

i +ysC is j. (3.5.11)

Theorem 3.5.4.The expression of h−tensor of deflection Dij and v−

tensor of deflection dij are given by

Dij = ysLi

s j−Nij , di

j = δ ij +ys

C

i

s j. (3.5.12)

Finally, we determine the horizontal paths, vertical pathsof DΓ (N)and its autoparallel curves.

The horizontal pathsof metrical connectionDΓ (N) = (Lijk,C

ijk)

are given by

d2xi

dt2+Li

jk

(x,

dxdt

)dxj

dtdxk

dt= 0, yi =

dxi

dt, (3.5.13)

where the coefficientsLijk are expressed by formula (3.5.2).

In the initial conditionsxi = xi0, yi =

(dxi

dt

)

0, local, this system of

differential equation has a unique solutionxi = xi(t), yi = yi(t), t ∈ I .Thevertical paths, at a point(xi

0) ∈ M are characterized by systemof differential equations:

dyi

dt+

C

i

jk(x0,y)yjyk = 0, xi = xi

0. (3.5.14)

In the initial conditionsxi = xi0,y

i = yi0 the previous system of differ-

ential equations, locally, has a unique solution.

3.7 The almost Hermitian model of the Lagrangian mechanicalsystemΣL 205

3.6 Electromagnetic field

For Lagrangian mechanical systemΣL = (M,L,Fe) the electromag-netic phenomena appear because the deflection tensorDi

j nonvan-ishes.

The covariant componentsDi j = gisDsj , di j = gisds

j determineh−e-lectromagnetic fieldFi j andv−electromagnetic fieldfi j as follows:

Fi j =12(Di j −D ji ), fi j =

12(di j −d ji ). (3.6.1)

Taking into account the formula (3.5.11) one obtains:

Fi j =F i j +

14

Fi j +14

Fi j (3.6.2)

whereF i j is the electromagnetic tensor field of the Lagrange space

Ln, Fi j is the elicoidal tensor field of mechanical systemΣL andFi j isthe skewsymmetric tensor

Fi j = ∂ jFsCirs − ∂iF

rCjrsys. (3.6.3)

In the case of Finslerian mechanical systemΣF the electromagnetic

tensorFi j is equal to14

Fi j .

Applying the method given in the chapter 2, part I, one gets theMaxwell equations of the Lagrangian mechanical systemΣL.

3.7 The almost Hermitian model of the Lagrangianmechanical systemΣL

Let N be the canonical nonlinear connection of the mechanical sys-temΣL and(δ i , ∂i) the adapted basis to the distributionN andV. Itsdual basis is(dxi ,δyi). TheN−lift G of fundamental tensorgi j on thevelocity spaceTM = TM\0 is given by

G= gi j dxi ⊗dxj +gi j δyi ⊗δy j . (3.7.1)


The almost complex structureF determined by the nonlinear con-nectionN is expressed by

F= δ i ⊗dxj − ∂i ⊗dxi . (3.7.2)

Thus, one proves:


1G is a pseudo-Riemannian structure andF is an almost complexstructure on the manifold TM. They depend only on the Lagrangianmechanical systemΣL.

2The pair(G,F) is an almost Hermitian structure.3The associated 2-form of(G,F) is given by

θ = gi j δyi ∧dxj .

4θ is an almost symplectic structure on the velocity spaceTM.5The following equality holds:

θ =θ −

14

Fi j dxi ∧dxj ,

θ being the symplectic structure of the Lagrange space Ln, i.e.

θ =

gi j

δyi ∧dxj .

Sinceθ is the symplectic structure of Lagrange spaceLn, we have

dθ = 0. So the exterior differentialdθ is as follows:

dθ =−14

dFi j ∧dxi ∧dxj . (3.7.3)

But dFi j = δ kFi j dxk+ ∂kFi j δyk. Consequently one obtain from (2.2):

dθ =−112

(Fi j |k+Fjk|i +Fki| j)dxk∧dxi ∧dxj −14

∂kFi j δyk∧dxi ∧dxj .

(3.7.4)Consequently:

1 θ =θ , if and only if the helicoidal tensorFi j vanish.

2 The almost symplectic structureθ of the Lagrangian mechanicalsystemΣL is integrable, if and only if the helicoidal tensorFi j satisfiesthe following tensorial equations:

3.8 Generalized Lagrangian mechanical systems 207

Fi j |k+Fjk|i +Fki| j = 0, ∂kFi j = 0, (3.7.5)

where the operator “|” is h−covariant derivation with respect to canon-ical N−linear connectionDΓ (N).

Now we observe that some good applications of this theory canbedone for the Lagrangian mechanical systemsΣL =(M,L(x,y),Fe(x,y))when the external forcesFeare of Liouville type:Fe= a(x,y)C.

3.8 Generalized Lagrangian mechanical systems

As we know, a generalized Lagrange spaceGLn = (M,gi j ) is givenby the configuration spaceM and by ad−tensor fieldgi j (x,y) on thevelocity spaceTM= TM\0, gi j being symmetric, nonsingular andof constant signature.

There are numerous examples of spacesGLn given by Miron R.[175], Anastasiei M. [14], R.G. Beil [40], [41], T. Kawaguchi [176]etc.

1 TheGLn space with the fundamental tensor

gi j (x,y) = α i j (x)+yiy j , y=α jyj (3.8.1)

andα i j (x) a semidefinite Riemann tensor.

2 gi j (x,y)=α i j (x)+

(1−

1n(x,y)

)yiy j , yi =α i j y j wheren(x,y)≥

1 is aC∞ function (the refractive index in Relativistic optics), [176].3 gi j (x,y)=α i j (x,y)+ayiy j , a∈R+, yi =α i j y j and whereα i j (x,y)

is the fundamental tensor of a Finsler space.The spaceGLn = (M,gi j (x,y)) is called reducible to a Lagrange

space if there exists a regular LagrangianL(x,y) such that

12

∂ 2L∂yi∂y j = gi j (x,y). (3.8.2)

A necessary condition that the spaceGLn = (M,gi j (x,y)) be re-ducible to a Lagrange space is that thed−tensor field

Ci jk =12

∂gi j

∂yk (3.8.3)


be totally symmetric.The following property holds:

Theorem 3.8.1.If we have a generalized Lagrange space GLn forwhich the fundamental tensor gi j (x,y) is 0-homogeneous with respectto yi and it is reducible to a Lagrange space Ln = (M,L), then thefunction L(x,y) is given by:

L(x,y) = gi j (x,y)yiy j +2Ai(x)y

i +U (x). (3.8.4)

The prove does not present some difficulties, [166].Also, we remark that in some conditions the fundamental tensor

gi j (x,y) of a spaceGLn determine a non linear connection, [241],[242].

For instance, in the cases 1,2,3 of the previous examples we

can take: 1,2,N

i

j =12

γ ijk(x)y

jyk, γ ijk(x) being the Christoffel sym-

bols of the Riemannian metricγ i j (x). For example 3 we takeNij =

12

γ ijk(x,y)y

jyk whereγ ijk(x,y) are the Christoffel symbols ofγ i j (x,y)

(see ch. 2, part I).Let us consider the functionE (x,y) on TM:

E (x,y) = gi j (x,y)yiy j (3.8.5)

called theabsolute energy[161] of the spaceGLn. The fundamen-tal tensorgi j (x,y) is say to be weakly regular if the absolute energyE (x,y) is a regular Lagrangian. That means: the tensor ˇgi j (x,y) =12

∂i ∂ jE is nonsingular. Thus the pair(M,E (x,y)) is a Lagrange space.

Definition 3.8.1.A generalized Lagrangian mechanical system is atriple ΣGL = (M,gi j ,Fe), whereGL = (M,gi j ) is a generalized La-grange space andFe is the vector field of external forces.Fe being avertical vector field we can write:

Fe= F i(x,y)∂ . (3.8.6)

In the following we assume that the fundamental tensorgi j (x,y) ofΣGL is weakly regular. Therefore we can give the following Postulate[49]:Postulate.The evolution equations (or Lagrange equations) of a gen-eralized Lagrange mechanical systemΣGL are:

3.8 Generalized Lagrangian mechanical systems 209

ddt

∂E

∂yi −∂E


dt(3.8.7)

with the covariant components ofFe:

Fi(x,y) = gi j (x,y)Fj(x,y). (3.8.8)

The Lagrange equations (3.8.7) are equivalent to the following sys-tem of second order differential equations:

d2xi

dt2+2Gi

(x,

dxdt

)=

12

F i

(x,

dxdt

)

2Gi =12

gis

(∂ 2E

∂ys∂x j yj −

∂E

∂xs

).

(3.8.9)

Therefore, one can apply the theory from this chapter for theLa-grangian mechanical systemΣL = (M,E (x,y),Fe). So we have:

Theorem 3.8.2.1The operator

S= yi∂i −2

(Gi −

12

F i)

∂i (3.8.10)

is a semispray on the velocity spaceTM.2The integral curves of S are the evolution curves ofΣGL.3 S is determined only by the generalized mechanical systemΣGL.

Theorem 3.8.3.The variation of energy EE = yi ∂E

∂yi −E are given by

dEE

dt=

dxi

dtFi

(x,

dxdt

). (3.8.11)

Feare called dissipative ifgi j F iy j ≤ 0.

Theorem 3.8.4.The energy EE is decreasing on the evolution curves(3.8.9) of systemΣGL if and only if the external forces Fe are dissipa-tive.

Other results:


4The canonical nonlinear connectionN of ΣG,L has the local coeffi-cients

Nij = ∂ j

G

i−

14

∂ jFi . (3.8.12)

5The canonicalN−linear metric connection of mechanical systemΣGL, DΓ (N) = (L jk,Cjk) has the coefficients:

Lijk =

12

gis

(δg js

δxk +δgsk

δx j −δg jk

δxs

)

Cijk =

12

gis

(∂g js

∂yk +∂gsk

∂y j −∂g jk

∂ys

).

(3.8.13)

By using the geometrical object fieldsgi j , S, N, DΓ (N) we canstudy the theory of Generalized Lagrangian mechanical systemsΣG,L.

Chapter 4Hamiltonian and Cartanian mechanical systems

The theory of Hamiltonian mechanical systems can be constructedstep by step following the theory of Lagrangian mechanical system,using the differential geometry of Hamiltonian spaces expound in partI. The legality of this theory is proved by means of Legendre dual-ity between Lagrange and Hamilton spaces. The Cartanian mechani-cal system appears as a particular case of the Hamiltonian mechani-cal systems. We develop here these theories using the author’s papers[154] and the book of R. Miron, D. Hrimiuc, H. Shimada and S. Sabau[174].

4.1 Hamilton spaces. Preliminaries

Let M be aC∞−real n−dimensional manifold, called configurationspace and(T∗M,π∗,M) be the cotangent bundle,T∗M is called mo-mentum (or phase) space. A pointu∗ = (x, p) ∈ T∗M, π∗(u∗) = x, hasthe local coordinate(xi , pi).

As we know from the previous chapter, a changing of local coordi-nate(x, p)→ (x, p) of pointu∗ is given by

xi = xi(x1, ...,xn), det

(∂ xi

∂x j

)6= 0

pi =∂x j

∂ xi p j

(4.1.1)

The tangent spaceT∗u∗T

∗M has a natural basis

(∂

∂xi = ∂i ,∂

∂ pi= ∂ i

)

with respect to (4.1.1) this basis is transformed as follows

211

212 4 Hamiltonian and Cartanian mechanical systems

∂∂xi =

∂ x j

∂xi

∂∂ x j +

∂ p j

∂xi

∂∂ p j

∂ i =∂xi

∂ x j˜∂

j(4.1.2)

Thus we have:1 On T∗M there are globally defined the Liouville 1-form:

p= pidxi (4.1.3)

and the natural symplectic structure

θ = dp= dpi ∧dxi . (4.1.4)

2 The vertical distributionV has a local basis(∂ 1, ..., ∂ n). It is ofdimensionn and is integrable.

3 A supplementary distributionN to the distributionV is given bya splitting:

Tu∗T∗M = Nu∗ ⊕Vu∗ , ∀u∗ ∈ T∗M. (4.1.5)

N is called a horizontal distribution or anonlinear connectionon themomentum spaceT∗M. The dimension ofN is n.

4 A local adapted basis toN andV are given by(δ i , ∂ i), (i =1, ...,n), with

δ i = ∂i +Nji ∂ j . (4.1.6)

The system of functionsNji (x, p) are the coefficients of the nonlinearconnectionN.

5ti j = Ni j −Nji (4.1.7)

is ad−tensor onT∗M - called the torsion tensor of the nonlinear con-nectionN. If ti j = 0 we say thatN is a symmetric nonlinear connec-tion.

6 The dual basis(dxi ,δ pi) of the adapted basis(δ i , ∂ i) has 1-formδ pi expressed by

δ pi = dpi −Ni j dxj . (4.1.8)

7

Proposition 4.1.1.If N is a symmetric nonlinear connection then the

symplectic structureθ can be written:

4.1 Hamilton spaces. Preliminaries 213

θ = δ pi ∧dxi . (4.1.9)

8 The integrability tensor of the horizontal distribution N is

Rki j = δ iNk j −δ jNki (4.1.10)

and the equations Rki j = 0 gives the necessary and sufficient conditionfor integrability of the distribution N.

The notion ofN−linear connection can be taken from the ch. 2, partI.

Definition 4.1.1.A Hamilton space is a pairHn = (M,H(x, p)) whereH(x, p) is a real scalar function on the momentum spaceT∗M havingthe following properties:

1H : T∗M →R is differentiable onT∗M = T∗M\0 and continuouson the null section of projectionπ∗.

2The Hessian ofH, with the elements

gi j =12

∂ i ∂ jH (4.1.11)

is nonsingular, i.e.

det(gi j ) 6= 0 onT∗M. (4.1.12)

3The 2-formgi j (x, p)η iη j has a constant signature onT∗M.

In the ch. 2, part I it is proved the following Miron’s result:

Theorem 4.1.1.1 In a Hamilton space Hn = (M,H(x, p)) for whichM is a paracompact manifold, there exist nonlinear connections de-termined only by the fundamental function H(x, p).

2 One of the,N, has the coefficients

Ni j =−

12

g jh

[14

gik∂ kH, ∂ hH+ ∂ h∂iH

]. (4.1.13)

3 The nonlinear connectionN is symmetric.

In the formula (4.1.13),, is the Poisson brackets.N is called the

canonical nonlinear connection ofHn.


The variational problem applied to the integral of action ofHn:

I(c) =∫ 1

0

[pi(t)

dxi

dt−H (x(t), p(t))

]dt, (4.1.14)

where

H (x, p) =12

H(x, p) (4.1.15)

leads to the following results:

Theorem 4.1.2.The necessary conditions as the functional I(c) bean extremal value of the functionals I(c) imply that the curve c(t) =(xi(t), pi(t)) is a solution of the Hamilton–Jacobi equations:

dxi

dt−

∂H

∂ pi= 0,

dpi

dt+

∂H

∂xi = 0. (4.1.16)

IfN is the canonical nonlinear connection then the Hamilton–

Jacobi equations can be written in the form

dxi

dt−

∂H

∂ pi= 0,

δ pi

dt+

δH

δxi = 0. (4.1.17)

Evidently, the equations (4.1.16) have a geometrical meaning.The curvesc(t) which verify the Hamilton–Jacobi equations are

called the extremal curves (or geodesics) ofHn.Other important results:

Theorem 4.1.3.The fundamental function H(x, p) of a Hamilton spaceHn = (M,H) is constant along to every extremal curves.


1 For a Hamilton space Hn there exists a vector field

ξ ∈X (T∗M)with the property

i ξ

θ =−dH. (4.1.18)

2

ξ is given by

ξ =12(∂ iH∂i −∂iH∂ i). (4.1.19)

4.2 The Hamiltonian mechanical systems 215

3 The integral curve of

ξ is given by the Hamilton–Jacobi equa-tions(4.1.16).

The vector field

ξ from the formula (4.1.19) is called the Hamiltonvector of the spaceHn.

Proposition 4.1.2.In the adapted basis of the canonical nonlinear

connectionN the Hamilton vector

ξ is expressed by

ξ =12(∂ iHδ i −δ iH∂ i). (4.1.20)

By means of the Theorem 4.1.4 we can say that the Hamilton vectorfield ξ is a dynamical system of the Hamilton spaceHn.

4.2 The Hamiltonian mechanical systems

Following the ideas from the first part, ch. ..., we can introduce thenext definition:

Definition 4.2.1.A Hamiltonian mechanical system is a triple:

ΣH = (M,H(x, p),Fe(x, p)), (4.2.1)

whereHn = (M,H(x, p)) is a Hamilton space and

Fe(x, p) = Fi(x, p)∂ i (4.2.2)

is a given vertical vector field on the momenta spaceT∗M.Fe is called the external forces field.

The evolution equations ofΣH can be defined by means of equa-tions (4.1.16) from the variational problem.

Postulate 4.2.1.The evolution equations of the Hamiltonian mechan-ical systemΣH are the followingHamilton equations:

dxi

dt− ∂ icalH = 0,

dpi

dt+∂iH =

12

Fi(x, p), H =12

H. (4.2.3)

Evidently, forFe= 0, the equations (4.2.3) give us the geodesics ofthe Hamilton spaceHn.


Using the canonical nonlinear connectionN we can write in an in-

variant form the Hamilton equations, which allow to prove the geo-metrical meaning of these equations.Examples. 1 ConsiderHn = (M,H(x, p)) the Hamilton spaces ofelectrodynamics, [19]:

H(x, p) =1

mcγ i j (x)pi p j −

2emc2Ai(x)pi +

e3

mc3Ai(x)Ai(x)

andFe= pi ∂ i . ThenΣH is a Hamiltonian mechanical system deter-mined only byHn.

2 Hn = (M,K2(x, p)) is a Cartan space andFe= pi ∂ i .3 Hn = (M,E (x, p)) with E (x, p) = γ i j (x)pi p j andFe= a(x)pi ∂ i .Returning to the general theory, we can prove:

Theorem 4.2.1.The following properties hold:1 ξ given by

ξ =12[∂ iH∂i − (∂iH −Fi)∂ i ] (4.2.4)

is a vector field onT∗M.2 ξ is determined only by the Hamiltonian mechanical systemΣH .3 The integral curves ofξ are given by the Hamilton equation

(4.2.3).

The previous Theorem is not difficult to prove if we remark thefollowing expression ofξ :

ξ =

ξ +12

Fe. (4.2.5)

Also we have:

Proposition 4.2.1.The variation of the Hamiltonian H(x, p) along theevolution curves ofΣH is given by:

dHdt

= Fidxi

dt. (4.2.6)

As we know, the external forcesFeare dissipative if〈Fw,C〉 ≥ 0.Looking at the formula (4.2.6) one can say:

4.3 Canonical nonlinear connection ofΣH 217

Proposition 4.2.2.The fundamental function H(x, p) of the Hamilto-nian mechanical systemΣH is decreasing on the evolution curves ofΣH , if and only if, the external forces Fe are dissipative.

The vector fieldξ onT∗M is called the canonical dynamical systemof the Hamilton mechanical systemΣH .

Therefore we can say: The geometry ofΣH is the geometry of pair(Hn,ξ ).

4.3 Canonical nonlinear connection ofΣH

The fundamental tensorgi j (x, p) of the Hamilton spaceHn is the fun-damental or metric tensor of the mechanical systemΣH . But othersfundamental geometric notions, as the canonical nonlinearconnec-tion of ΣH cannot be introduced in a straightforward manner. Theywill be defined by means ofL−duality between the Lagrangian andthe Hamiltonian mechanical systemsΣL andΣH (see ch. ..., part I).

Let ΣL = (M,L(x,y),Fe1(x,y)), Fe

1= F

1i(x,y)∂i be a Lagrangian me-

chanical system. The mapping

ϕ : (x,y) ∈ TM → (x, p) ∈ T∗M, pi =12

∂iL

is a local diffeomorphism. It is called theLegendre transformation.Let ψ be the inverse ofϕ and

H(x, p) = 2piyi −L(x,y), y= ψ(x, p). (4.3.1)

One prove thatHn = (M,H(x, p)) is an Hamilton space. It is theL−dual of Lagrange spaceLn = (M,L(x,y)).

One proves thatϕ transform:

1 The canonical semispraySof Ln in the Hamilton vector

ξ of Hn.

2 The canonical nonlinear connectionNL of Ln into the canonical

nonlinear connectionNH of Hn.

3 The external forcesFe1

of ΣL into external forcesFeof ΣH , with

Fi(x, p) = gi j (x, p)Fj

1 (x,ψ(x, p)).4 The canonical nonlinear connectionNL of ΣL with coefficients

into the canonical nonlinear connectionNH of ΣH with the coefficients


Ni j (x, p) =Ni j (x, p)+

14

gih∂ hFj . (4.3.2)

Ni j are given by (...., ch. ...).

Therefore we have:

Proposition 4.3.1.The canonical nonlinear connection N of the Hamil-tonian mechanical systemΣH has the coefficients Ni j , (4.3.2).

Of course, directly we can prove thatNi j are the coefficients of anonlinear connection. It is canonical forΣH , sinceN depend only onthe mechanical systemΣH .

The torsion ofN is

ti j =14(gih∂ hFj −g jh∂ hFi). (4.3.3)

Evidently∂ iFj = 0, implies thatN is symmetric nonlinear connectionon the momenta spaceT∗M.

Let (δ i , ∂ i) be the adapted basis toN and V and (dxi ,δ pi) itsadapted cobasis:

δ i = ∂i +Nji ∂ j = δ i +14

g jh∂ hFi ∂ j ,

δ pi = dpi −Ni j dxj =

δ pi −14

gih∂ hFidxj .

(4.3.4)

The tensor of integrability of the canonical nonlinear connectionN is

Rki j = δ iNk j −δ jNki. (4.3.5)

Rki j = 0 characterize the integrability of the horizontal distributionN.

The canonicalN−metrical connectionCΓ (N) = (H ijk,C

jki ) of the

Hamiltonian mechanical systemΣH is given by the following Theo-rem:

Theorem 4.3.1.The following properties hold:1) There exists only one N−linear connection CΓ = (Ni j ,H i

jk,Cjki )

which depend on the Hamiltonian systemΣH and satisfies the axioms:1 Ni j from (4.3.2) is the canonical nonlinear connection.2 CΓ is h− metric:

gi j|k = 0. (4.3.6)

4.4 The Cartan mechanical systems 219

3 CΓ is v−metric:gi j |k = 0. (4.3.6’)

4 CΓ is h−torsion free:

T ijk = H i

jk −H ik j = 0. (4.3.7)

5 CΓ is v−torsion free

Sjki =C jk

i −Ck ji = 0. (4.3.7’)

2) The coefficients of CΓ are given by the generalized Christoffelsymbols:

H ijk =

12

gis(δ jgsk+δ kg js−δ sg jk)

C jki =−

12

gis(∂ jgsk+ ∂ kg js− ∂ sg jk)

(4.3.8)

Now, we are in possession of all data to construct the geometryof Hamilton mechanical systemΣH . Such that we can investigate theelectromagnetic and gravitational fields ofΣH .

4.4 The Cartan mechanical systems

An important class of systemsΣH is obtained when the HamiltonianH(x, p) is 2-homogeneous with respect to momentapi .

Definition 4.4.1.A Cartan mechanical system is a set

ΣC = (M,K(x, p),Fe(x, p)) (4.4.1)

whereC n = (M,K(x, p)) is a Cartan space and

Fe= Fi(x, p)∂ i (4.4.2)

are the external forces.

The fact thatC n is a Cartan spaces implies:1 K(x, p) is a positive scalar function onT∗M.2 K(x, p) is a positive 1-homogeneous with respect to momentapi .3 The pairHn = (M,K2(x, p)) is a Hamilton space.


Consequently:a. The fundamental tensorgi j (x, p) of C n is

gi j =12

∂ i ∂ jK2. (4.4.3)

b. We haveK2 = gi j pi p j . (4.4.4)

c. The Cartan tensor is

Ci jk =14

∂ i ∂ j ∂ kK2, piCi jk = 0.

d. C n = (M,K(x, p)) is L− dual of a Finsler spaceFn = (M,F(x,y)).

e. The canonical nonlinear connectionN, established by R. Miron

[175], has the coefficients

Ni j= γh

i j ph−12(γh

srphpr)∂ sgi j , (4.4.5)

γ ijk(x, p) being the Christoffel symbols ofgi j (x, p).Clearly, the geometry ofΣC is obtained from the geometry ofΣH

takingH = K2(x, p).So, we have:

Postulate 4.4.1.The evolution equations of the Cartan mechanicalsystemΣC are the Hamilton equations:

dxi

dt−

12

∂ iK2 = 0,dpi

dt+

12

∂iK2 =

12

Fi(x, p). (4.4.6)

A first result is given by

Proposition 4.4.1.1 The energy of the Hamiltonian K2 is given byEK2 = pi ∂ iK2−K2 = K2.

2 The variation of energyEK2 = K2 along to every evolution curve(4.4.6) is

dK2

dt= Fi

dxi

dt. (4.4.7)

Example. ΣC , with K(x, p) = γ i j (x)pi p j1/2, (M,γ i j (x)) being a

Riemann spaces andFe= a(x, p)pi ∂ i .

4.4 The Cartan mechanical systems 221

Theorem ... of part ... can be particularized in:

Theorem 4.4.1.The following properties hold good:1 ξ given by

ξ =12[∂ iK2∂i − (∂iK

2−Fi)∂ i ] (4.4.8)

is a vector field onT∗M.2 ξ is determined only by the Cartan mechanical systemΣC .3 The integral curves ofξ are given by the evolution equations

(4.4.6) of ΣC .

The vectorξ is Hamiltonian vector field onT∗M or of the Cartanmechanical systemΣC .

Therefore, the geometry ofΣC is the differential geometry of thepair (C n,ξ ).

The fundamental object fields of this geometry areC n,ξ ,Fe, thecanonical nonlinear connectionN with the coefficients

Ni j =Ni j +

14

gih∂ hFj , (4.4.9)

Taking into account that the vector fieldsδ i = ∂i −Nji ∂ j determine anadapted basis to the nonlinear connectionN, we can get the canonicalN− metrical connectionCΓ (N) of ΣC .

Theorem 4.4.2.The canonical N− metrical connection CΓ (N) of theCartan mechanical systemΣC has the coefficients

H ijk =

12

gis(δ jgsk+δ kg js−δ sg jk), C jki = gisC

s jk. (4.4.10)

Using the canonical connectionsN andCΓ (N) one can study theelectromagnetic and gravitational fields on the momenta space T∗Mof the Cartan mechanical systemsΣC , as well as the dynamical systemξ of ΣC .

Chapter 5Lagrangian, Finslerian and Hamiltonianmechanical systems of orderk≥ 1

The notion of Lagrange or Finsler spaces of higher order allows us tointroduce the Lagrangian and Finslerian mechanical systems of orderk≥ 1. They will be defined as an natural extension of the Lagrangianand Finslerian mechanical systems. Now, the geometrical theory ofthese analytical mechanics is constructed by means of the differentialgeometry of the manifoldTkM−space of accelerations of orderk andof the notion ofk−semispray, presented in the part II.

5.1 Lagrangian Mechanical systems of orderk≥ 1

We repeat shortly some fundamental notion about the differential ge-ometry of the acceleration spaceTkM.

Let TkM be the acceleration space of orderk (see ch. 1, part II)andu ∈ TkM, u = (x,y(1), ...,y(k)) a point with the local coordinates(xi ,y(1)i , ...,y(k)i).

A smooth curvec : I → M represented on a local chartU ⊂ M byxi = xi(t), t ∈ I can be extended to(πk)−1(U )⊂ TkM by c : I → TkM,given as follows:

xi = xi(t), y(1)i =11!

dxi(t)dt

, ...,y(k)i =1k!

dkxi

dtk(t), t ∈ I . (5.1.1)

Consider the vertical distributionV1,V2, ...,Vk. They are integrableand have the propertiesV1 ⊃V2 ⊃ ...⊃Vk. The dimension ofVk is n,dimVk−1 = 2n, ..., dimV1 = kn.

The Liouville vector fields onTkM are:

223

224 5 Lagrangian, Finslerian and Hamiltonian mechanical systems of orderk≥ 1

1Γ = y(1)i

∂∂y(k)i

2Γ = y(1)i

∂∂y(k−1)i

+2y(2)i∂

∂y(k)i.............kΓ = y(1)i

∂∂y(1)i

+2y(2)i∂

∂y(2)i+ ...+ky(k)i

∂∂y(k)i

.

(5.1.2)

Thus1Γ ∈Vk,

2Γ ∈Vk−1, ...,

kΓ ∈V1.

The following operator:

Γ = y(1)i∂

∂xi +2y(2)i∂

∂y(1)i+ ...+ky(k)i

∂∂y(k−1)i

(5.1.3)

is not a vector field onTk <, but it is frequently used. OnTkM thereexists ak tangent structureJ defined in ch. 2.J is integrable and hasthe properties

ImJ =V, kerJ =Vk, rankJ = kn (5.1.4)

andJ(1Γ ) = 0, J(

2Γ ) =

1Γ , ...,J(

kΓ ) =

k−1Γ , JJ ...J= 0 (k times).

A k−semispray is a vector fieldS on the acceleration spaceTkMwith the property

JS=kΓ . (5.1.5)

Also, S is called adynamical systemon TkM.The local expression of ak−semisprayS is (ch. 1, part II)

S= Γ − (k+1)Gi ∂∂y(k)i

. (5.1.6)

The system of functionsGi(x,y(1), ...,y(k)) are thecoefficientsof S.The integral curves of thek−semispraySare given by the following

system of differential equations

dk−1xi

dtk−1 +(k+1)Gi(x,y(1), ...,y(k)) = 0

y(1)i =dxi

dt, ...,y(k)i =

1k!

dkxi

dtk.

(5.1.7)

5.1 Lagrangian Mechanical systems of orderk≥ 1 225

If Fe= F i(x,y(1), ...,y(k))∂

∂y(k)iis an arbitrary vertical vector field

into the vertical distributionVk andS is ak−semispray, thus

S′ = S+Fe (5.1.8)

is ak−semispray, becauseJS′ = JS=kΓ .

Recall the notion of nonlinear connection (ch. 1, part II):A nonlinear connection on the manifoldTkM is a distributionN

supplementary to the vertical distributionV =V1:

TuTkM = N(u)+V(u), ∀u∈ TkM. (5.1.9)

An adapted basis to the distributionN is (ch. 1, part II)

δδxi =

∂∂xi −N

1

j

i

∂∂y(1) j

− ...−Nk

j

i

∂∂y(k) j

. (5.1.10)

The systems of functionsN1

j

i, ...,N

k

j

iare the coefficients of nonlinear

connectionN.Let N1, ...,Nk−1,Vk the vertical distributions defined by

N1= J(N), N2= J(N1), ...,Nk−1= J(Nk−2), Vk= J(Nk−1) andN0=N.(5.1.11)

Thus, dimN0 =dimN1 = ...=dimNk = n and we have

TuTkN = N0(u)⊕N1(u)⊕ ...⊕Nk−1(u)⊕Vk, ∀u∈ TkM. (5.1.12)

The adapted basis to the direct decomposition (5.1.12) is asfollows:(

δδxi ,

δδy(1)i

, ...,δ

δy(k)i

), (5.1.13)

whereδ

δxi is given by (5.1.10) and

δδy(1)i

= J

(δ

δxi

),

δδy(2)i

= J

(δ

δy(1)i

), ...,

δδy(k)i

=

= J

(δ

δy(k−1)i

)=

∂∂y(k)i

.

(5.1.14)


The expressions of the basis (5.1.13) can be easily obtainedfrom theformulas (5.1.10), (5.1.14) (ch. 1, part II).

The dual basis of the adapted basis (5.1.13) is:

(δxi ,δy(1)i , ...,δy(k)i), (5.1.15)

where

δxi = dxi

δy(1)i = dy(1)i +M1

i

jdxj

δy(2)i = dy(2)i +M1

i

jdy(1) j +M

2i

jdxj

.......................δy(k)i = dy(k)i +M

1i

jdy(k−1) j + ...+M

k

i

jdxj

(5.1.16)

and the systems of functionsM1

i

j, ...,M

k

i

jare the dual coefficients of

the nonlinear connectionN. The relations between dual coefficientsM1

i

j, ...,M

k

i

jand (primal) coefficientsN

1

i

j, ...,N

k

i

jare (ch. 1, part II):

M1

i

j= N

1i

j,

M2

i

j= N

2

i

j+N

1

m

jM1

i

m,

..................................Mk

i

j= N

k

i

j+ N

k−1

m

jM1

i

m+ ...+N

1m

jN

k−1

i

m.

(5.1.17)

This recurrence formulas allow us to calculate the primal coefficientsN1

i

j, ...,N

k

i

jfunctions by the dual coefficientsM

1i

j, ...,N

k

i

j.

In the adapted basis (5.1.13) the Liouville vector fields1Γ , ...,

kΓ can

be expressed:

5.2 Lagrangian mechanical system of orderk, ΣLk 227

1Γ = z(1)i

δδy(k)i

2Γ = z(1)i

δδy(k−1)i

+2z(2)iδ

δy(k)i................................kΓ = z(1)i

δδy(1)i

+2z(2)iδ

δy(2)i+ ...+kz(k)i

δδy(k)i

,

(5.1.18)

wherez(1)i , ...,z(k)i arethe d-Liouville vector fields:

z(1)i = y(1)i

2z(2)i = 2y(2)i +M1

i

my(1)m

................................

kz(k)i = ky(k)i +(k−1)M(1)

i

m

y(k−1)m+ ...+(k−1)M(1)

i

m

y(1)m.

(5.1.19)

To a change of local coordinates (ch. 1, part II) on the accelerationmanifoldTkM, every one of the Liouvilled−vectors transform on therule:

z(α)i =∂ xi

∂x j z(α) j , (α = 1, ...,k). (5.1.20)

So, they have a geometric meaning.

5.2 Lagrangian mechanical system of orderk, ΣLk

Definition 5.2.1.A Lagrangian mechanical system of orderk≥ 1 is atriple:

ΣLk = (M,L(x,y(1), ...,y(k)),Fe(x,y(1), ...,y(k))), (5.2.1)

whereL(k)n = (M,L(x,y(1), ...,y(k))) (5.2.2)

is a Lagrange space of orderk≥ 1, and

Fe(x,y(1), ...,y(k)) = F i(x,y(1), ...,y(k))∂

∂y(k)i(5.2.3)


is an a priori given vertical vector field.Clearly,Fe is a vector field belonging to the vertical distributionVk

and its contravariant componentsF i is a distinguished vector field.Feor F i are called the external forces ofΣLn.

Applying the theory of Lagrange spaces of orderk≥ 1 to the spaceL(k)n we have the fundamental (or metric) tensor field ofΣLk:

gi j =12

∂ 2L

∂y(k)i∂y(k) j, (5.2.4)

and the Euler–Lagrange equations

Ei(L)

det=

∂L∂xi −

ddt

∂L

∂y(1)i+ ...+(−1)k dk

dtk∂L

∂y(k)i= 0

y(1)i =dxi

dt, ...,y(k)1 =

1k!

dkxi

dtk.

(5.2.5)

This is a system of differential equations of order 2k, autoadjoint. Itallows us to determine a canonical semispray of orderk on TkM.

Indeed, sinceEi(L) is ad−covector field onTkM, we can calculate

E(φ(t)L(x,y(1),...,y(k))) and obtain (ch. 1, part II):

Ei(φL) = φ

Ei +

dφdt

1Ei(L)+ ...+

dkφdtk

kEi(L). (5.2.6)

The coefficients1Ei(L), ...,

kEi(L) ared−covector fields discovered by

Craig and Synge (we refer to books [161]).So, we have:

k−1E i(L) = (−1)k−1 1

(k−1)!

[∂L

∂y(k−1)i−

ddt

∂L

∂y(k)i

]. (5.2.7)

Theorem 5.2.1.For any Lagrange spaces L(k)n =(M,L) the followingproperties hold:

1 the system of differential equations gi jk−1E i(L) = 0 has the form

dk+1xi

dtk+1 +(k+1)!G

i(x,

dxdt

, ...,1k!

dkx

dtk

)= 0 (5.2.8)

5.3 Canonicalk−semispray of mechanical systemΣLk 229

with

(k+1)G

i=

12

gi j[

Γ∂L

∂y(k) j−

∂L

∂y(k−1) j

]. (5.2.9)

2 The system of functionsG

ifrom (5.2.9) are the coefficients of

k−semisprayS which depends only on the Lagrangian L(x,y(1), ...,

y(k)).

So,Sis thecanonical semisprayof L(k)n. But it leads to a canonical

nonlinear connectionN for L(k)n.

Indeed, we have

Theorem 5.2.2 (Miron [161]).S being the canonical k−semispray of

Lagrange space L(k)n, there exists a nonlinear connectionN of the

space L(k)n which depend only on the fundamental function L. The

non-connectionN has the dual coefficients:

M1

i

j=

∂Gi

∂y(k) j,

M2

i

j=

12(SM

1i

j+M

1i

mM1

m

j),

........................................

Mk

i

j=

1k(S M

k−1

i

j+ M

k−1

i

mM

k−1

m

j).

(5.2.10)

Now, we can develop the geometry of Lagrange spaceL(k)n by

means of the geometrical object fieldsL,S,

N andgi j (ch. 2, part II).

We apply these considerations to the geometrization of Lagrange me-chanical systemΣLk.

5.3 Canonicalk−semispray of mechanical systemΣLk

The fundamental equations of mechanical systemΣk are given by:Postulate 5.3.1.The evolution equations of the Lagrangian mechan-ical system of order k,ΣLk = (M,L,Fe) are the following Lagrangeequations:


ddt

∂L

∂y(k)i−

∂L

∂y(k−1)i= Fi,

y(1)i =dxi

dt, ...,y(k)i =

1k!

dkxi

dtk,

(5.3.1)

where

Fi(x,y(1), ...,y(k)) = gi j (x,y

(1), ...,y(k))F i(x,y(1), ...,y(k)) (5.3.2)

are the covariant components of external forces Fe.Evidently, in the casek=1, the previous equations are the evolution

equations of Lagrangian mechanical systemΣL =(M,L(x,y),Fe(x,y)).Also, the equation (5.3.1) fork = 1 are the Lagrange equations forRiemannian or Finslerian mechanical systems. These arguments jus-tify the introduction of previous Postulate.

Examples.1 Let L(k)n = (M,L(x,y(1), ...,y(k))) be a Lagrange space

of order k ≥ 1, andz(k)i

be its Liouville d−vector field of orderk.Consider the external forces

Fe= hz(k)i ∂

∂y(k)i, h∈ R∗. (5.3.3)

Thus,ΣLk = (M,L(x,y(1), ...,y(k))), Fe(x,y(1), ...,y(k)) is a Lagrangianmechanical system of orderk.

The covariant componentFi of Fe is Fi = hgi jz(k) j

, which substitutein (5.3.1) give us the Lagrange equations of orderk for the consideredmechanical system.

In the general case of mechanical systemsΣLk is important to provethe following property:

Theorem 5.3.1.The Lagrange equations(5.3.1) are equivalent to thefollowing system of differential equations of order k+1:

dk+1xi

dtk+1 +(k+1)!G

i(x,

dxdt

, ...,1k

dkx

dtk

)=

k!2

F i (5.3.4)

with the coefficientsG

i(x,y(1), ...,y(k)) from (5.2.9).

Proof. We have

5.3 Canonicalk−semispray of mechanical systemΣLk 231

ddt

∂L

∂y(k)i−

∂L

∂y(k−1)i= Γ

∂L

∂y(k)i−

∂L

∂y(k−1)i+

2k!

gi jdk+1x j

dtk+1 −Fi .

It follows

dk+1xi

dtk+1 +k!2

gi j(

Γ∂L

∂y(k) j−

∂L

∂y(k−1) j

)=

k!2

F i . (5.3.4’)

But the previous equations is exactly (5.3.4), with the coefficients(k+

1)G

igiven by (5.2.9).

In initial conditions:(xi)0 = xi0,

(dxi

dt

)

0= y(1)i0 , ...,

1k!

(dkxi

dtk

)

0=

y(k)i0 , locally there exists a unique solution of evolution equations(5.3.4):xi = xi(t), y(1)i = y(1)i(t), ...,y(k)i = y(k)i(t), t ∈ (a,b) whichexpress the moving of the mechanical systemΣLk.

But, it is convenient to write thek−Lagrange equations (5.3.4) inthe form

y(1)i =dxi

dt, y(2)i =

12

d2xi

dt2, ...,y(k)i =

1k!

dkxi

dtk

dy(k)i

dt+(k+1)

G

i(x,y(1), ...,y(k)) =

12

F i(x,y(1), ...y(k)).

(5.3.5)

These equations determine the trajectories of the vector field Son theacceleration spaceTkM:

S= y(1)i∂

∂xi +2y(2)i∂

∂y(1)i+ ...+ky(k)i

∂∂y(k−1)i

−

−(k+1)G

i ∂∂y(k)i

+12

F i ∂∂y(k)i

(5.3.6)

or

S=S+

12

Fe (5.3.6’)

whereS is the canonicalk−semispray of the Lagrange space of order

k, L(k)n = (M,L). Consequently, we have:


Theorem 5.3.2.1 For the Lagrangian mechanical system of order k,ΣLk the operator S from(5.3.6) is a k−semispray which depend onlyon the mechanical systemΣLk.

2 The integral curves of S are the evolution curves given by La-grange equations(5.3.5).

S is calledthe canonical semispray ofΣLk or it is called thedynam-ical system ofΣLk, too.

The coefficientsGi of the canonicalk−semispraySare:

(k+1)Gi = (k+1)G

i−

12

F i. (5.3.7)

In the example (5.3.3) the canonicalk−semispray has the coeffi-cients

(k+1)Gi = (k+1)G

i−

h2

z(k)i . (5.3.8)

Of course, the vector fieldS on the manifold of the acceleration oforderk ≥ 1, TkM being a dynamical system it can be used for inves-tigate the qualitative problems, as stability of solutions, equilibriumpoints etc.

Now, we can study the canonical nonlinear connection of the me-chanical systemΣLk.

5.4 Canonical nonlinear connection of mechanicalsystemΣLk

Let Gi be the coefficients of the canonical semispraySof mechanicalsystemΣLk. Thus

Gi =G

i−

12(k+1)

F i , (5.4.1)

whereG

i=

12(k+1)

gi j[

Γ∂L

∂y(k) j−

∂L

∂y(k−1) j

]. (5.4.2)

The dual coefficientsM1

i

j, ...,

Mk

i

jof the canonical nonlinear connection

N of Lagrange spaceL(k)n = (M,L) are given by (5.2.10):

5.4 Canonical nonlinear connection of mechanical systemΣLk 233

M1

i

j=

∂G

i

∂y(k) j,M2

i

j=

12(S

M1

i

j+

M1

i

m

M1

m

j), ...,

Mk

i

j=

1k(S

M

k−1

i

j+

M1

i

m

M

k−1

m

j).

(5.4.3)

By virtue of (5.3.6’):

S=S+

12

Fe (5.4.4)

the dual coefficientsM1

i

j, ...,M

k

i

jof canonical nonlinear connectionN

of ΣLk can be written:

M1

i

j=

M1

i

j−

12(k+1)

∂F i

∂y(k+1) j

M2

i

j=

12(SM

1i

j+M

1i

mM1

m

j)

........................

M(k)

i

j

=1k(S M

k−1i

j+M

1i

mM

k−1m

j)

(5.4.5)

where

SMα

i

j=

SM

αi

j+

12

Fe(Mα

i

j), α = 1,2, ...,k−1. (5.4.6)

The coefficientsN1

i

j, ..., N

(k)

i

j

of N are given by the formulas

N1

i

j= M

1

i

j, N(α)

i

j

= Mα

i

j− N

(α−1)

m

j

M(1)

i

m

− ...− N(1)

m

j

M(α−1)

i

m

, (α = 2, ...,k).

(5.4.7)

The adapted basisδ

δxi (i = 1, ...,n) to the distributionN and the

adapted basis to the vertical distributionsV1, ...,Vk:(

δδxi ,

δδy(1)i

, ...,δ

δy(k)i

)(5.4.8)

can be explicitly written.


Its dual adapted basis

(δxi ,δy(1)i , ...,δy(k)i) (5.4.9)

has the following expression:

δxi = dxi , δy(1)i = dy(1)i +M(1)

i

j

dxj , ...,δy(k)i = dy(k)i+

+M(1)

i

j

dy(k−1) j + ...+M(k)

i

j

dxj .(5.4.9’)

By using the Definition from part II, equations (1.5.9), we have

Theorem 5.4.1.The autoparallel curves of the canonical nonlinearconnection N of the mechanical systemΣLk are given by the followingsystem of differential equation

δy(1)i

dt= 0, ...,

δy(k)i

dt= 0

dxi

dt= y(1)i , ...,

1k!

dkxi

dtk= y(k)i .

(5.4.10)

Theorem 5.4.2.The variation of the Lagrangian L along the autopar-allel curves of the canonical nonlinear connection N are given by

dLdt

=δLδxi

dxi

dt. (5.4.11)

Corollary 5.4.1. The Lagrangian L of the space L(k)n = (M,L) of amechanical systemΣLk is conserved along the autoparallel curves of

the canonical nonlinear connection N iff the scalar productδLδxi

dxi

dtvanishes.

Remark 5.4.1.The Bucataru’s nonlinear connectionbN of the mechan-

ical systemΣLk has the coefficients

5.5 Canonical metricalN−connection 235

bN(1)

i

j

=∂

G

i

∂y(k) j+

12(k+1)

∂F i

∂y(k) j

N(2)

i

j

=∂

G

i

∂y(k−1) j+

12(k+1)

∂F i

∂y(k−1) j

.........................

N(k)

i

j

=∂

G

i

∂y(1) j+

12(k+1)

∂F i

∂y(1) j.

(5.4.12)

This nonlinear connection depend only on the mechanical systemΣLk,too. SometimeN

bis more convenient to solve the problems of the ge-

ometrization of mechanical systemΣLk.

5.5 Canonical metricalN−connection

Let N be the canonical nonlinear connection with the coefficients

(5.4.7) orN being the Bucataru’s connectionbN with the coefficients

(5.4.12). AnN−linear connectionD with the coefficientsDΓ (N) =(Li

jh, C(1)

i

jh

, ...,C(k)

i

jh

) is metrical with respect to the metric tensorgi j of

the mechanical systemΣLk (cf. ch. 2, part II) if the following equationshold:

gi j |h = 0, gi j

(α)

| h = 0, (α = 1, ...,k). (5.5.1)

Thus, the theorem ... implies:Theorem 5.5.1.The we have:(I) There exists a unique N−linear connection D onTkM satisfyingthe axioms:

(1) N is the canonical nonlinear connection of the Lagrange spaceL(k)n (or N is Bucataru nonlinear connection);

(2) gi j |h = 0, (D is h−metrical);

(3) gi j

(α)

| h = 0, (α = 1, ...,k) (D is vα−metrical);(4) T i

jh = 0 (D is h−torsion free);

(5) S(α)

i

jh

= 0 (D is vα−torsion free).


(II) The coefficients CΓ (N) = (Lhi j , C

(1)

h

i j

, ..., C(k)

h

i j

) of D are given by the

generalized Christoffel symbols

Lhi j =

12

ghs

(δgis

δx j +δgs j

δxi −δgi j

δxs

)

Chi j

(α)

=12

ghs

(δgis

δy(α) j+

δgs j

δy(α)i−

δgi j

δy(α)s

), (α = 1, ...,k).

(III ) D depends only on the mechanical systemΣLk.

TheN−metrical connectionD is called thecanonical metrical con-nectionof the Lagrangian mechanical system of orderk≥ 1 ΣLk.

Now, one can affirm:the geometrization ofΣLk can be developedby means of the canonical semispray S and by the geometrical objectfields: L,gi j ,N, and CΓ (N).

5.6 The Riemannian(k−1)n almost contact model ofthe Lagrangian mechanical system of orderk, ΣLk

N being the nonlinear connection with the coefficients (5.4.7) (or withthe coefficients (5.4.12)) and(δxi ,δy(1)i , ...,δy(k)i) be the adaptedcobasis to the direct decomposition (5.1.12).

Consider theN−lift G of the metric tensorgi j :

G= gi j dxi ⊗dxj +gi j δy(1)i ⊗δy(1) j + ...+gi j δy(k) j ⊗δy(k)i . (5.6.1)

Thus,G is a semidefinite Riemannian structure on the manifoldTkMdepending only onΣLk.

The distributionsN0,N

1, ..., N

k−1,Vk are two by two orthogonal with

respect toG.The nonlinear connectionN uniquely determines the(k−1)n−almost

contact structureF by:

F

(δ

δxi

)=−

∂∂y(k)i

; F

(δ

δy(α)i

)=0, α =1, ...,k−1; F

(∂

∂y(k)i

)=

δδxi .

(5.6.2)Indeed, it follows:

5.6 The Riemannian(k−1)n almost contact model of the Lagrangian mechanical system oforderk, ΣLk237

Theorem 5.6.1.(1)F is globally defined on the manifoldTkM anddepend onΣLk, only.

(2)kerF= N1⊕N2⊕ ...⊕Nk−2, ImF= N0⊕Vk.(3)rank‖F‖= 2n.(4)F3+F= 0.

So,F is an almost(k−1)n−contact structure on the bundle of ac-celeration of order k, TkM.

Let

(ξ1a, ξ2a, ..., ξ

(k−1)a

), a = 1, ...,n, be a local basis adapted to the

direct decompositionN1⊕ ...⊕ N

k−1and

1aη ,

2aη , ...,

(k−1)aη its dual basis.

Thus the set(F, ξ

1a, ..., ξ

(k−1)a,1aη , ...,

(k−1)aη

),(a= 1, ...,n) (5.6.3)

is a(k−1)n−almost contact structure.Indeed, we have

F

(ξαa

)= 0,

a aη

(ξβ b

)=

δ a

b, for α = β0, for α 6= β , (α,β = 1, ...,n−1)

(5.6.4)and

F2(X) =−X+n

∑a=1

k−1

∑α=1

a aη (X)ξ

αa, ∀X ∈ X (TkM)

a aη F= 0.

(5.6.5)

The Nijenhuis tensorNF is expressed by

NF(X,Y) = [FX,FY]+F2(X,Y)−F[FX,Y]−F[X,FY]. (5.6.6)

The structure (5.6.5) isnormal if:

NF(X,Y)+n

∑a=1

k−1

∑α=1

da aη (X,Y) = 0, ∀X,Y ∈ X (TkM). (5.6.7)


A characterization of the normality of structure (4.6.7) isas follows:

Theorem 5.6.2.The almost(k−1)n−contact structure

(F, ξ

αa,a bη)

is

normal, if and only if, for any vector fields X,Y onTkM we have

NF(X,Y)+n

∑a=1

k−1

∑α=1

d(y(α)a)(X,Y) = 0. (5.6.8)

Thus, we can prove:

Theorem 5.6.3.The pair (G,F) is a Riemannian(k− 1)n−almost

contact structure onTkM, depending only on the Lagrangian mechan-ical system of order k≥ 1, ΣLk.

Indeed, the following condition holds:

G(FX,Y) =−G(FY,X), ∀X,Y ∈ X (TkM). (5.6.9)

Therefore, the triple(TkM,G,F) is called the Riemannian(k−1)n−almost contact model of the mechanical systemΣLk. It can be used tosolve the problem of geometrization of mechanical systemΣLk.

5.7 Classical Riemannian mechanical system withexternal forces depending on higher orderaccelerations

Let us consider the classical Riemannian mechanical systems

ΣR = (M,T,Fe), (5.7.1)

with T =12

γ i j (x)dxi

dtdxj

dtas kinetic energy and the external forces de-

pending on the acceleration of order 1,2, ...,k, i.e.Fe(x,y(1), ...,y(k)).The problem iswhat kind of evolution equations can be postulate inorder to study the movings of these systems?

Of course, in nature we have such kind of mechanical systems.Forinstance, the friction forces, which action on a rocket in the time ofits moving are the external forces depending by higher orderacceler-ations. But, it is clear that the classical Lagrange equations:

5.7 Classical Riemannian mechanical system with external forces depending on higher order accelerations239

ddt

(∂ (2T(x,y(1)))

∂y(1)i

)−

∂ (2T(x,y(1)))∂xi = Fi(x,y

((1)), ...,y(1))

have the form

d2xi

dt2+ γ i

jh(z)dxj

dtdxh

dt=

12

γ is(x)Fs

(x,

dxdt

, ...,1k!

dkx

dtk

)

and fork> 2 it is not a differential system of equations of order 2, likein the classical Lagrange equations.

We solve this problem by means of the prolongation of orderk ofthe Riemannian spaceRn = (M,γ i j (x)).

Let ΣR = (M,2T,Fe) a mechanical system of form (5.7.1) with

T =12

γ i j (x)y(1)iy(1) j ,

(y(1)i =

dxi

dt

)as kinetic energy. Thus the Rie-

mannian spaceRn = (M,γ i j (x)) can be prolonged to the Riemannianspace of orderk≥ 1 ProlkRn = (TkM,G), where

G= γ i j (x)δxi ⊗δx j +γ i j (x)δy(1)i⊗δy(1) j + ...+γ i j (x)δy(k)i⊗δ y(k) j .(5.7.2)

The 1-formsδxi ,δy(1)i , ...,δy(k)i being determined as follows:

δxi = dxi , δy(1)i = dy(1)i +M(1)

i

j

dxj , ...,δy(k)i = dy(k)i+

+M(1)

i

j

dy(k−1)i + ...+Mk

i

jdxj .

(5.7.3)

In these formulasM(α)

i

j

(α = 1, ...,k) are the dual coefficients of a

nonlinear connectionN determined only on the Riemannian structure

γ i j (x). That are


M1

i

j(x,y(1)) = γ i

jh(x)y(1)m,

M2

i

j(x,y(1),y(2)) =

12

(Γ M

(1)

i

j

+M(1)

i

m

M(1)

m

j

)

..................................

Mk

i

j(x,y(1), ...,y(k)) =

12

(Γ M

(k−1)

i

j

+M(1)

i

m

M(k−1)

m

j

),

(5.7.4)

Γ being the nonlinear operator:

Γ = y(1)i∂

∂xi +2y(2)i∂

∂y(1)i+ ...+ky(k)i

∂∂y(k−1)i

. (5.7.5)

Thus, (cf. ch. ...) the dual coefficientsM(α)

(5.7.4) and the primal coef-

ficients N(α)

(5.4.7) depend on the Riemannian structureγ i j (x), only.

Now the Lagrangian mechanical system of orderk:

ΣProlkRn = (M,L(x,y(1), ...,y(k))), Fe(x,y(1),y(k)),

L = γ i j (x)z(k)iz(k) j ,

kz(k)i = ky(k)i +M(1)

i

m

y(k−1)m+ ...+ M(k−1)

i

m

y(1)m,

(5.7.6)

depend only onΣRn. It is the prolongation of orderk of the Rieman-nian mechanical system (4.7.1),ΣRn.

We can postulate:The Lagrange equations ofΣRn are the Lagrange equations of the

prolonged Lagrange equations of order k,ΣProlkRn:

ddt

(∂L(x,y(1), ...,y(k))

∂y(k)i

)−

∂L(x,y(1), ...,y(k))

∂y(k−1)i= Fi(x,y(1), ...,y(k))

y(1)i =dxi

dt, ...,y(k)i =

1k!

dkxi

dtk.

(5.7.7)Applying the general theory from this chapter, we have

Theorem 5.7.1.The Lagrange equations of the classical RiemanniansystemsΣR with external forces depending on higher order accelera-

5.8 Finslerian mechanical systems of orderk, ΣFk 241

tions are given by

dk+1xi

dtk+1 +(k+1)!G

i(x,y(1), ...,y(k)) =

12

k!F i(x,y(1), ...,y(k))

(k+1)G

i=

12

γ i j

[Γ

∂L

∂y(k) j−

∂L

∂y(k−1)

]

L = γ i j (x)z(k)iz(k) j .

(5.7.8)

The canonical semispray ofΣR is expressed by

S= y(1)i∂

∂xi + ...+ky(k)i∂

∂y(k−1)i− (k+1)

G

i ∂∂y(k)i

+12

F i ∂∂y(k)i

.

(5.7.9)The integral curves ofSare given by the Lagrange equations (5.7.15).

The canonical semisprayS is the dynamical system of the classicalLagrangian mechanical systemΣR with the external forces dependingon the higher order accelerations.

5.8 Finslerian mechanical systems of orderk, ΣFk

The theory of the Finslerian mechanical systems of orderk is a naturalparticularization of that of Lagrangian mechanical systems of orderk≥ 1, described in the previous sections of the present chapter.

Definition 5.8.1.A Finslerian mechanical system of orderk ≥ 1 is atriple:

ΣFk = (M,F(x,y(1), ...,y(k))),Fe(x,y(1), ...,y(k)), (5.8.1)

whereF(k)n = (M,F(x,y(1), ...,y(k))) (5.8.2)

is a Finsler space of orderk≥ 1, and

Fe(x,y(1), ...,y(k)) = F i(x,y(1), ...,y(k))∂

∂y(k)i(5.8.3)

is an a priori given vertical vector field.

F(x,y(1), ...y(k)) is the fundamental function ofΣFk,


gi j =12

∂∂y(k)i

∂∂y(k) j

F2 (5.8.4)

is the fundamental tensor ofΣFk andFeare the external forces.

The Euler–Lagrange equations ofF(k)n are given by (5.2.5)Ei(F2)=

0, y(1)i =dxi

dt, ...,y(k)i =

1k!

dkxi

dtkand the Craig–Synge covector field is

Ek−1(F2) from (5.2.7). Theorem 5.2.1 forL = F2 can be applied.

Thus, the canonical semispraySof F (k)n is:

S= y(1)i

∂∂xi +2y(2)i

∂∂y(1)i

+ ...+ky(k)i∂

∂y(k−1)i− (k+1)

G

∂∂y(k)i

,

(5.8.5)with the coefficients:

(k+1)G

i=

12

gi j[

Γ∂F2

∂y(k)i−

∂F2

∂y(k−1)i

](5.8.6)

and with nonlinear operator

Γ = y(1)i∂

∂xi +2y(2)i∂

∂y(1)i+ ...+ky(k)i

∂∂y(k−1)i

. (5.8.7)

The canonical nonlinear connectionN of F(k)n has dual coefficients

(5.2.10) particularized in the following form:

M1

i

j=

∂G

i

∂y(k) j

M2

i

j=

12(S

M1

i

j+

M1

i

m

M1

m

j),

.................................Mk

i

j=

1k(S

M

k−1

i

j+

M

k−1

i

m

M

k−1

m

j).

(5.8.8)

The fundamental equations of the Finslerian mechanical systems ofordern≥ 1, ΣFn are given by

Postulate 4.8.1.The evolution equations of the Finslerian mechanicalsystemΣFk = (M,F,Fe) are the following Lagrange equations

5.8 Finslerian mechanical systems of orderk, ΣFk 243

ddt

∂F2

∂y(k)i−

∂F2

∂y(k−1)i= Fi , Fi = gi j F j ,

y(1)i =dxi

dt, ...,y(k)i =

1k!

dkxi

dtk.

(5.8.9)

Remark 5.8.1.Fork= 1, (5.8.8) are the Lagrange equations of a Fins-lerian mechanical systemΣF = (M,F,Fe).

The Lagrange equations are equivalent to the following system ofdifferential equations of orderk+1:

dk+1xi

dtk+1 +k!2

gi j[

Γ∂F2

∂y(k)i−

∂F2

∂y(k−1)

]=

k!2

F i . (5.8.10)

Integrating this system in initial conditions we obtain an unique so-lution xi = xi(t), which express the moving of Finslerian mechanicalsystemΣFk.

But it is convenient to write the system (5.8.9) in the form (5.3.5):

y(1)i =dxi

dt, y(2)i =

12

d2xi

dt2, ...,y(k)i =

1k!

dkxi

dtk

dy(k)i

dt+(k+1)

G

i(x,y(1), ...,y(k)) =

12

F i(x,y(1), ...,y(k)).

(5.8.11)

These equations determine the trajectories of the vector field

S=S+

12

Fe. (5.8.12)

Consequently, we have:

Theorem 5.8.1.1 For the Finslerian mechanical system of order k≥1, ΣFk the operator S from(5.8.12) is a k−semispray which dependonly onΣFk.

2 The integral curves of S are the evolution curves ofΣFk.

S is the canonical semispray of the mechanical systemΣFk. It isnamed the dynamical system ofΣFn, too.

The geometrical theory of the Finslerian mechanical systemΣFk

can be developed by means of the canonical semisprayS, applyingthe theory from the sections 5.4, 5.5, 5.6 from this chapter.


Remark 5.8.2.Let ΣF = (M,F,Fe) be a Finslerian mechanical system,whereFe depend on the higher order accelerations. What kind of La-grange equations we have forΣF?

The problem can be solved by means of the prolongation of FinslerspaceFn = (M,F) to the manifoldTkM, following the method usedin the section 5.7 of the present chapter.

5.9 Hamiltonian mechanical systems of orderk≥ 1

The Hamiltonian mechanical systems of orderk ≥ 1 can be studiedas a natural extension of that of Hamiltonian mechanical systems oforderk= 1, given byΣH = (M,H(x, p),Fe(x, p)), ch. 4, part II.

So, a Hamiltonian mechanical system of orderk≥ 1 is a triple

ΣHk = (M,H(x,y(1), ...,y(k−1)), p),Fe(x,y(1), ...,y(k−1), p) (5.9.1)

whereH(k)n = (M,H(x,y(1), ...,y(k−1)), p) (5.9.2)

is a Hamilton space of orderk≥ 1 (see definition 5.1.1) and

Fe(,y(1), ...,y(k−1), p) = Fi(x,y(1), ...,y(k−1), p)∂ i (5.9.3)

are the external forces.

Of course,∂ i =∂

∂ pi.

Looking to the Hamilton–Jacobi equation of the spaceH(k)n we canformulate

Postulate 5.9.1.The Hamilton equations of the Hamilton mechanicalsystemΣHk are as follows:

5.9 Hamiltonian mechanical systems of orderk≥ 1 245

H =12

H

dxi

dt=

∂H

∂ pi

dpi

dt=−

∂H

∂xi+

ddt

∂H

∂y(1)i+ ...+(−1)

1(k−1)!

ddt

dk−1

dtk−1

∂H

∂yk−1 =12

Fi

y(1)i =dxi

dt, y(2)i =

12

d2xi

dt2, ...,y(k−1) =

1(k−1)!

dk−1xi

dtk−1 .

(5.9.4)It is not difficult to see that these equations have a geometric mean-

ing.For k = 1 the equations (5.9.4) are the Hamilton equations of a

Hamiltonian mechanical systemΣH = (M,H(x, p),Fe(x, p)) .These reasons allow us to say that the Hamilton equations (5.9.4)

are the fundamental equationsfor the evolution of the Hamiltonianmechanical systems of orderk≥ 1.

Example 5.9.1.The mechanical systemΣHk with

H(x,y(1), ...,y(k−1), p) = αγ i j (x,y(1), ...,y(k−1))pi p j , (5.9.5)

γ i j being a Riemannian metric tensor on the manifoldT(k−1)n andwith

Fe= β i(x,y(1), ...,y(k−1))∂ i (5.9.5’)

β i is ad−covector field onTk−1M andα > 0 a constant.Let us consider the energy of orderk−1 of

E k−1(H) = I k−1(H)−12!

ddt

I k−2(H)+ ...+

+(−1)k−2 1(k−1)!

dk−2

dtk−2(H)−H,

(5.9.6)

whereI (1)H = L 1

ΓH, ..., I k−1H = Lk−1

Γ.

Theorem 5.9.1.The variation of energyE k−1(H) along the evolutioncurves(5.9.4) can be calculate without difficulties.


Thus, the canonical nonlinear connectionN∗ of ΣHk is given by thedirect decomposition (5.1.11), and has the coefficientsN

(1)

i

j

,..., N(k−1)

i

j

,

Ni j from (5.1.12’) and its dual coefficientsM(1)

,..., M(k−1)

from (5.1.14).

We can calculate the Liouville vector fieldsz(1), ...,z(k−1) from (5.1.17).The metricalN∗−linear connection has the coefficientsDΓ (N∗) =

(Hsi j , C

(k)

i

jk

,C jsi ) given by (5.5.2).

Now we can say: The geometries of the Hamiltonian mechanicalsystems of orderk ≥ 1, ΣHk can be constructed only by the funda-mental equations (5.9.4), by the nonlinear connection ofΣHk and bytheN∗−metrical connectionCΓ (N∗).

Remark 5.9.1.The homogeneous case of Cartanian mechanical sys-tems of orderk, ΣC k = (M,K(x,y(1), ...,y(k−1)), p), Fe(x,y(1), ...,y(k−1)p) is a direct particularization of the previous theory forH(x,y(1),..., y(k−1), p) = K2(x,y(1), ..., y(k−1), p), whereK(x,y(1), ...,y(k−1), p)is k−homogeneous with respect to variables(y(1), ...,y(k−1), p).

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