Lagrangian Mechanics on Lie Algebroids

LAGRANGIAN MECHANICS ON LIE ALGEBROIDS

EDUARDO MARTINEZ

Abstract. A geometric description of Lagrangian Mechanics on Liealgebroids is developed in a parallel way to the usual formalism ofLagrangian Mechanics on the tangent bundle of a manifold. Thedynamical system defined by a Lagrangian is shown to be symplecticin a generalized sense.

Contents

1. Introduction 22. Lie algebroids 33. The bundle LE 64. The Lie algebroid structure of LE 105. The Liouville section and the vertical endomorphism 156. Second-order differential equations 187. Lagrangian formalism 198. Noether’s theorem 239. Examples 2410. Conclusions and outlook 28References 28

1. Introduction

The concept of Lie algebroid is a generalization of both the conceptof a Lie algebra and the concept of an integrable distribution. In a re-cent paper, Weinstein [10] develops a generalized theory of LagrangianMechanics on Lie algebroids. Examples of Lagrangian systems on Liealgebroids are, among others, systems defined on Lie algebras, systemswith symmetries on principal fiber bundles, systems on semidirect prod-ucts and systems with holonomic constraints.

1

2 EDUARDO MARTINEZ

The equations of motion were found by means of the pullback of thecanonical Poisson structure on the dual of the algebroid, when the La-grangian is regular. Weinstein asks the question of whether it is possibleto develop a formalism similar to Klein’s formalism [4] in ordinary La-grangian Mechanics, which allows a direct construction of the equationof motion without reference to the structures on the dual. Later, Liber-mann [5] considers that question and shows that such formalism is notpossible, in general, if we consider the tangent bundle TE to the Liealgebroid τ : E → M as the space for developing the theory, that is, asthe substitute of T (TM) in the usual formalism, when E = TM .

The aim of this paper is to provide such a formalism. From the workof Libermann, we are obliged to develop our theory in a space LE →E which in general is not TE but reduces to it whenever E = TM .Therefore vectorfields and differential forms are substituted by sectionsof this bundle and its dual.

The fundamental objects needed to develop the Lagrangian formalismare the Liouville vectorfield, the vertical endomorphism and the exteriordifferential(see [2]). The Liouville vectorfield exists in any vector bundleand this (among other reasons) forces us to choose LE to be a vectorbundle over E close to TE. The vertical endomorphism maps, roughlyspeaking, horizontal directions to vertical directions. Therefore it willexists only on vector bundles of even rank equal to twice the rank of Eover M . This fact forces the choice of LE. Finally, in order to have anexterior differential operator we will need to provide to LE with a Liealgebroid structure.

The paper is organized as follows. In section 2 we recall some basicfacts about Lie algebroids and the differential geometry associated tothem. In section 3 we define a bundle LE which plays the role of TTMin the usual formulation on Lagrangian Mechanics and in section 4 weendow such bundle with a natural Lie algebroid structure. In section 5we study two fundamental geometric objects defined on LE which arethe equivalents of the vertical endomorphism and the Liouville vector-field on TM , and we show that much of the properties of this objects arepreserved in this generalization. In section 6 we define the analog of asecond order differential equation on a manifold. In section 7 we definethe Cartan sections and we prove that, for regular Lagrangians, Hamiltonequations defined by the energy function by using the Cartan section assymplectic form are the Euler-Lagrange equations. As an application westate in section 8 two versions of Noether’s theorem, one generalizing the

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LAGRANGIAN MECHANICS ON LIE ALGEBROIDS 3

classical Noether’s Theorem for point transformations, and a general ver-sion which admits a converse. Some illustrative examples are presentedin section 9.

2. Lie algebroids

We consider a vector bundle τ : E → M . A structure of Lie algebroidon E is given by a Lie algebra structure on the C∞(M)-module of sectionsof the bundle, (Sec(E), [ , ]), together with a homomorphism ρ : E → TMof vector bundles which induces a Lie algebra homomorphism (denotedwith the same symbol) ρ : Sec(E)→ X(M), satisfying the compatibilitycondition

[σ1, fσ2] = f [σ1, σ2] + ρ(σ1)f σ2.

where f is a smooth function on M and σ1, σ2 are sections of E.Therefore, we also have the relations

[ρ(σ1), ρ(σ2)] = ρ([σ1, σ2]).

and[σ1, [σ2, σ3]] + [σ2, [σ3, σ1]] + [σ3, [σ1, σ2]] = 0,

for σ1, σ2 and σ3 sections of E. Examples of Lie algebroids are TM , anintegrable distribution of TM , the Atiyah algebroid of a principal fiberbundle, and M × g when the Lie algebra g acts on the manifold M .See [10] for the details.

Is is useful to think of a Lie algebroid E over M as a new tangentbundle for M . Sections of E plays the role of vectorfields on the manifoldM . Similarly, the algebra

∧(E) = Sec((E∗)∧p → M) of multilinear

alternating forms on E plays the role of the algebra of differential formsfor M . The Lie algebroid properties enable one to define an exteriordifferential operator d on

∧(E) as follows. If f is a function on M , then

we define df(m) ∈ E∗m by

〈 df(m) , a 〉 = ρ(a)f, for every a ∈ Em.If θ is an element of

∧p(E) with p > 0, then we define the element dθ of∧p+1(E) by the formula

dθ(σ1, . . . , σp+1) =

p+1∑i=1

(−1)i+1ρ(σi)θ(σ1, . . . , σi, . . . , σp+1)

+∑i<j

(−1)i+jθ([σi, σj], σ1, . . . , σi, . . . , σj, . . . , σp+1).

where the hat over an argument means the absence of that argument.

4 EDUARDO MARTINEZ

For instance, if θ is a section of E∗ we have

dθ(σ1, σ2) = ρ(σ1)〈 θ , σ2 〉 − ρ(σ2)〈 θ , σ1 〉 − 〈 θ , [σ1, σ2] 〉.It is easy to see that d2 = 0. It can be seen that the existence of anexterior differential on

∧(E) is equivalent to a structure of Lie algebroid

on E.Throughout this work d will represent the differential on the Lie al-

gebroid and should not be confused with the exterior differential on amanifold.

If V is an element of Xr(E) = Sec((E∗)∧p ⊗ E → M) we can definethe operator dV mapping

∧p(E) to∧p+r(E) by means of

dV θ = iV dθ + (−1)rdiV θ,

where iV is the inner contraction with V . See [9] for the details. Inparticular, if σ is a section of τ we obtain an operator dσ which plays therole of the Lie derivative

dσθ = iσdθ + diσθ.

The usual property d ◦ dσ = dσ ◦ d holds, as well as the relations

dσiη − iηdσ = i[σ,η] and dσdη − dηdσ = d[σ,η].

Note that over functions f on M we have dσf = ρ(σ)f .

A function f on M can be lifted to a function f on E by pull-back

f(a) = f(τ(a)) for a ∈ E.

A section θ of the dual bundle π : E∗ → M also defines a function θ onE by means of

θ(a) = 〈 θm , a 〉 for a ∈ Em.A function of this kind will be called a linear function. When θ is thedifferential of a function f on M the corresponding linear function willbe denoted by f . Therefore

f = df .

Notice that not every linear function is a linear combination of functionsof the form f with coefficients functions on M . This holds only when theanchor ρ is injective.

If E is finite dimensional, we take local coordinates (xi) on M and alocal base {eα} of sections of the bundle. Then we have local coordinates(xi, yα) on E, where yα(a) is the α-th coordinate of a ∈ E in the givenbase. Such coordinates determine local functions ρiα, Cα

βγ on M which

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contains the local information of the Lie algebroid structure, and accord-ingly they are called the structure functions of the Lie algebroid. Theyare given by

ρ(eα) = ρiα∂

∂xiand [eα, eβ] = Cγ

αβeγ.

These functions should satisfy the relations

ρjα∂ρiβ∂xj− ρjβ

∂ρiα∂xj

= ρiγCγαβ,

and ∑ciclic(α,β,γ)

[ρiα∂Cν

βγ

∂xi+ Cµ

ανCνβγ

]= 0

which are usually called the structure equations.In local coordinates the differential d is determined by

dxi = ρiαeα and deα = −1

2Cαβγe

β ∧ eγ,

where {eα} is the dual base of {eα}. Note that the structure equationsgiven above are but d2xi = 0 and d2eα = 0.

The differential of a function f on M has the local expression

df =∂f

∂xiρiαe

α.

If θ = θαeα is a section of E∗ then the linear function θ is

θ(x, y) = θαyα.

It follows that the function f is of the form

f(x, y) =∂f

∂xiρiαy

α.

In particular xi = ρiαyα.

If we change coordinates x i = x i(x) on the base manifold M and linearcoordinates y α = Aαβy

β on E, corresponding to a new base {e α} givenby eβ = Aαβ e α, then the transformation rule of the structure functionsare

ρiα = ρ jβAβα

∂xi

∂x j

CγαβA

µγ = C µ

γνAγαA

νβ + ρiα

∂Aµβ∂xi− ρiβ

∂Aµα∂xi

.

6 EDUARDO MARTINEZ

3. The bundle LE

In this section we will define a bundle LE over E, which we call theprolongation of E, and we will study some canonical lifting procedures ofsections of τ . The bundle LE plays the role of τTM : T (TM) → TM inthe ordinary Lagrangian Mechanics. In the next section we will provideLE with a Lie algebroid structure.

The total space of the prolongation is the total space of the pull-backof Tτ : TE → TM by the anchor map ρ,

LE = { (b, v) ∈ E × TE | ρ(b) = Tτ(v) } ,

but fibered over E by the projection τ1 : LE → E, given by τ1(b, v) =τE(v), where τE : TE → E is the tangent projection. For clarity in theexposition we will use the (redundant) notation (a, b, v) to denote theelement (b, v) of LE, where a ∈ E is the point where v is tangent. Withthis notation

LE = { (a, b, v) ∈ E × E × TE | τ(a) = τ(b), v ∈ TaE and ρ(b) = Taτ(v) }

and the bundle projection is

τ1(a, b, v) = a.

The sum and product by real numbers are then expressed as

(a, b1, v1) + (a, b2, v2) = (a, b1 + b2, v1 + v2)

λ(a, b, v) = (a, λb, λv).

The other natural projections are also important in the theory. Wedefine the projection τ2 : LE → E as the projection onto the secondfactor, τ2(a, b, v) = b. The map τ2 plays the role of the projectionTτM : TTM → TM . The projection onto the third factor ρ1 : LE → TE,ρ1(a, b, v) = v, will be the anchor of the prolonged algebroid, as we will seein the next section. Finally we define the projection τ12 : LE → E×M Eas the projection onto the first two factors, τ12(a, b, v) = (a, b).

An element of LE is said to be vertical if it is in the kernel of theprojection τ2. Therefore it is of the form (a, 0, v) with v a vertical vectortangent to E at a. The set of vertical elements in LE is a vector subbun-dle of LE and will be denoted by Ver(LE). If z is vertical, then ρ1(z)is a vertical vector on E. But it is important to note that if ρ1(z) is avertical vector, then z could be non-vertical. This only holds when ρ isan injective map.


Being E a vector bundle, the fibers can be identified with the verticaltangent spaces via the vertical lift b 7→ bVa , defined by

bVaF =d

dtF (a+ tb)

∣∣t=0,

for an arbitrary function F on E.This allows us to define the vertical lifting map ξV : E ×M E → LE

given by ξV (a, b) = (a, 0, bVa ), which is a vector bundle isomorphism frompr1 : E ×M E → E to τ1 : Ver(LE) → E. If σ is a section of τ thenthe section σV of τ1 defined by σV (a) = ξV (a, σ(τ(a))) will be called thevertical lift of σ.

The following properties of the vertical lift of a section are easy toprove:

ρ1(σV )f = 0 ρ1(σV )θ = iσθ,

for a function f on M and a section θ of E∗.

A section η of LE is said to be projectable if there exists a section σ ofτ such that τ2 ◦ η = σ ◦ τ . In such case we will say that η is a lifting of σ.As before, it is important to note that this is not equivalent to the vectorfield ρ1(η) being projectable to M : if a section η is projectable then ρ1(η)is a projectable vector field on E, but the converse is (in general) false.The set of projectable sections will be denoted by Secpr(LE).

Is is also important to note that (in general) an element z = (a, b, v)of LE is not determined by the action of v on functions. Instead, z canbe defined by the element (a, b) of E ×M E to which projects and bythe action of v = ρ1(z) on linear functions. Of course, this action mustsatisfy the compatibility property

v(f θ) = θ(a) ρ(b)f + f(a) v(θ).

This allows us to define the complete lift of a section as follows.

Theorem: Given a section σ ∈ Sec(τ) there exists one and only onesection σC ∈ Sec(τ1) that projects to σ and satisfies

ρ1(σC)(θ) = dσθ,

for every section θ of E∗. The section σC will be called the complete liftof σ.

Proof. We just have to prove that the given action on linear functions is

consistent. Therefore, we consider the map v : θ 7→ dσθ. If f is a functionon the base M , then

dσ(fθ) = (dσf θ + fdσθ)∧ = dσf θ + f dσθ.

8 EDUARDO MARTINEZ

Thus

v(f θ) = dσf θ + f v(θ),

and the result follows by noticing that dσf = ρ(σ)f .

From the definition it follows that

ρ1(σC)f = dσf ρ1(σC)θ = dσθ,

for a function f on M and a section θ of E∗.

Proposition: The complete and vertical lift satisfy the properties

(fσ)V = fσV and (fσ)C = fσC + fσV ,

for f a function on M and σ a section of E.

Proof. The first one is a consequence of the linearity of ξV . For thesecond, we first note that both sections projects to fσ. Therefore if θ isa section of E∗ we have

ρ1((fσ)C)θ = dfσθ

= (fdσθ + df iσθ)∧

= f dσθ + df iσθ

= fρ1(σC)θ + fρ1(σV )θ,

where we have used the definition of f and ρ1(σV )θ = iσθ.

Assume that E is finite dimensional and consider a local base {eα} ofsections of τ , so that we have coordinates (xi, yα) on E. Then we havelocal coordinates (xi, yα, zα, vα) on LE given as follows. If (a, b, v) is anelement of LE and has coordinates (mi, aα) for a, (mi, bα) for b, then v

is of the form v = ρiαbα ∂∂xi

∣∣∣a

+ vα ∂∂yα

∣∣∣a. The coordinates of (a, b, v) are

(mi, aα, bα, vα). It follows that the coordinate expression of the map ρ1,considered as a vector field along τ1, is

ρ1(x, y, z, v) = ρiαzα ∂

∂xi

∣∣∣(x,y)

+ vα∂

∂yα

∣∣∣(x,y)

.


The local base {Xα,Vα} of sections of LE associated to the coordinatesystem is given by

Xα(a) =

(a, eα(τ(a)), ρiα

∂

∂xi

∣∣∣a

)Vα(a) =

(a, 0,

∂

∂yα

∣∣∣a

).

If V is a section of LE which in coordinates reads

V (x, y) = (xi, yα, Zα(x, y), V α(x, y)),

then the expression of V in terms of base {Xα,Vα} is

V = ZαXα + V αVα.

and the vector field ρ1(V ) ∈ X(E) has the expression

ρ1(V ) = ρiαZα(x, y)

∂

∂xi

∣∣∣(x,y)

+ V α(x, y)∂

∂yα

∣∣∣(x,y)

.

The expressions of the vertical lift of a section σ = σαeα and thecorresponding vector field are

σV = σαVα, and ρ1(σV ) = σα∂

∂yα.

The expression of the complete lift of a section σ is

σC = σαXα + (σα − Cαβγσ

βyγ)Vα,

and therefore

ρ1(σC) = ρiασα ∂

∂xi+ (σα − Cα

βγσβyγ)

∂

∂yγ.

Under a change of coordinates x i = x i(x), y α = Aαβyβ on E the

transformation rule of the coordinates on LE is

x i = x i(x)

y α = Aαβyβ

z α = Aαβzβ

v α = Aαβvβ + ρiβ

∂Aαγ∂xi

zβyγ

10 EDUARDO MARTINEZ

and the corresponding equations of the change of base are

Xβ = Aαβ X α + ρiβ∂Aαγ∂xi

yγ V α

Vβ = Aαβ V α.

From here it immediately follows that the equations of change of the dualbasis {X α,Vα} are

X α = Aαβ X β

V α = Aαβ Vβ + ρiβ∂Aαγ∂xi

yγ X β.

4. The Lie algebroid structure of LE

In this section we will endow LE with a Lie algebroid structure. Weuse the fact that the set of vertical and complete lifts of sections of Eis a generating set of Sec(LE), and therefore we can define the bracketonly for this kind of vectors, and declaring ρ1 to be the anchor.

Theorem: There exists one and only one Lie algebroid structure onτ1 : LE → E such that the anchor is ρ1 and the bracket [ , ] satisfies therelations

[σV , ηV ] = 0

[σV , ηC] = [σ, η]V

[σC, ηC] = [σ, η]C,

for σ, η ∈ Sec(E).

Proof. We first prove that the given relations for complete a vertical liftsare consistent with the anchor ρ1.

If we multiply a section η by a function f ∈ C∞(M) then the definitionsays

[σC, (fη)C] = [σ, fη]C

= (ρ(σ)f η + f [σ, η])C

= ˜ρ(σ)fηC +˙

ρ(σ)fηV + f [σ, η]C + f [σ, η]V

On the other hand, if we apply first the rule (fη)C = fηC + fηV we have

[σC, (fη)C] = [σC, fηC + fηV ]

= ρ1(σC)f ηC + f [σC, ηC] + ρ1(σC)f ηV + f [σC, ηV ]


which coincides with the former expression by virtue of the relationsbetween ρ and ρ1 given in the last section.

Similarly, by definition

[(fσ)C, ηV ] = [fσ, η]V

= (−ρ(η)f σ + f [σ, η])V

= −˜ρ(η)fσV + f [σ, η]V ,

and on the other hand

[(fσ)C, ηV ] = [fσC + fσV , ηV ]

= −ρ1(ηV )f σC + f [σC, ηV ]− ρ1(ηV )f σV + f [σV , ηV ]

= f [σ, η]V −˜ρ(η)f σV .

Now if we take the vertical lift of fη, the definition states

[σC, (fη)V ] = ([σ, fη])V

= ρ(σ)f η + f [σ, η]V

= ˜ρ(σ)fηV + f [σ, η]V ,


[σC, (fη)V ] = [σC, fηV ]

= ρ1(σC)fηV + f [σC, ηV ]

= ρ1(σC)fηV + f [σ, η]V .

Finally, from the definition

[σV , (fη)V ] = 0


[σV , (fη)V ] = [σV , fηV ]

= ρ1(σV )fηV + f [σV , ηV ]

= 0.

Now we prove that the bracket satisfies the Jacobi identity. For threevertical lifts we have

[σV1 , [σV

2 , σV

3 ]] + [σV2 , [σV

3 , σV

1 ]] + [σV3 , [σV

1 , σV

2 ]] = 0 + 0 + 0 = 0.

12 EDUARDO MARTINEZ

For two vertical lifts and a complete lift we have

[σV1 , [σV

2 , σC

3 ]]+[σV2 , [σC

3 , σV

1 ]] + [σC3 , [σV

1 , σV

2 ]] =

= [σV1 , [σ2, σ3]V ] + [σV2 , [σ3, σ1]V ] + 0 = 0 + 0 = 0

For two complete lifts and a vertical lift we have

[σV1 , [σC

2 , σC

3 ]] + [σC2 , [σC

3 , σV

1 ]] + [σC3 , [σV

1 , σC

2 ]] =

= [σV1 , [σ2, σ3]C] + [σC2 , [σ3, σ1]V ] + [σC3 , [σ1, σ2]V ]

= [σ1, [σ2, σ3]]V + [σ2, [σ3, σ1]]V + [σ3, [σ1, σ2]]V = 0.

And finally for three complete lifts we have

[σC1 , [σC

2 , σC

3 ]] + [σC2 , [σC

3 , σC

1 ]] + [σC3 , [σC

1 , σC

2 ]] =

= [σC1 , [σ2, σ3]C] + [σC2 , [σ3, σ1]C] + [σC3 , [σ1, σ2]C]

= [σ1, [σ2, σ3]]C + [σ2, [σ3, σ1]]C + [σ3, [σ1, σ2]]C = 0.

To end the proof we have to show that the anchor ρ1 is a Lie algebrahomomorphism. To begin with, for two vertical lifts it is clear that[ρ1(σV ), ρ1(ηV )] = 0 = ρ1([σV , ηV ]).

For a vertical and a complete lift on basic functions we have

[ρ1(σC), ρ1(ηV )]f = ρ1(σC)ρ1(ηV )f − ρ1(ηV )ρ1(σC)f

= −ρ1(ηV )˜ρ(σ)f

= 0

and

ρ1([σC, ηV ])f = ρ1([σ, η]V )f

= 0

The action on linear functions is

[ρ1(σC), ρ1(ηV )]θ = ρ1(σC)ρ1(ηV )θ − ρ1(ηV )ρ1(σC)θ

= ρ1(σC)iηθ − ρ1(ηV )dσθ

= dσiηθ − iηdσθ

= i[σ,η]


and

ρ1([σC, ηV ])θ = ρ1([σ, η]V )θ

= i[σ,η]

Finally, for two complete lifts the action on basic functions is

[ρ1(σC), ρ1(ηC)]f = ρ1(σC)ρ1(ηC)f − ρ1(ηC)ρ1(σC)f

= ρ1(σC)˜ρ(η)f − ρ1(ηC)˜ρ(σ)f

= ˜ρ(σ)ρ(η)f − ˜ρ(η)ρ(η)f

= ˜[ρ(σ), ρ(η)]f

and

ρ1([σC, ηC])f = ρ1([σ, η]C)f

= ˜ρ([σ, η])f

= ˜[ρ(σ), ρ(η)]f

The action on linear functions is

[ρ1(σC), ρ1(ηC)]θ = ρ1(σC)ρ1(ηC)θ − ρ1(ηC)ρ1(σC)θ

= ρ1(σC)dηθ − ρ1(ηV )dσθ

= dσdηθ − dηdσθ

= d[σ,η]θ


ρ1([σC, ηC])θ = ρ1([σ, η]C)θ

= d[σ,η]θ

This completes the proof.

Remark: Notice that, in general, there can be other algebroid structureson LE with the same bracket. Indeed, if k is linear map k : E → E suchthat =(k) ⊂ Ker ρ then ρ1 = ρ1 + ξV ◦ k ◦ τ2 is compatible with thegiven bracket if and only if [k(σ), η] + [σ, k(η)] = k([σ, η]), for every pairof sections σ and η of E. This is a consequence of the fact that thefunctions of the form f and f do not span the algebra of functions on Ewhen ρ is not injective.

14 EDUARDO MARTINEZ

Remark: Janusz Grabowski has pointed out that the Lie algebroid LEis a pull-back of E. Indeed, it can be shown that LE = τ ∗∗E, the inducedLie algebroid by the projection map τ : E → M (see [6]). Nevertheless,the definition given above is more adequate for our purposes.

In terms of the differential d on LE we have the properties

dσC f = dσf dσV f = 0

dσC θ = dσθ dσV θ = iσθ.

which are in fact equivalent to the definition of complete and vertical lift.

Proposition: If X and Y are projectable sections of LE, then [X, Y ]is projectable and

τ2([X,Y ]) = [τ2(X), τ2(Y )],

where we are simplifying the notation by writing τ2(X) for the section σof E such that τ2 ◦X = σ ◦ τ .

Proof. We first prove that the bracket of two vertical sections is vertical.Indeed, let V and W vertical sections. Then we can write V =

∑A VAσ

VA

andW =∑

AWAσVA for some sections σA of E and some functions VA,WA

on E. Therefore

[V,W ] =∑A,B

VAWB[σVA, σV

B] +∑A

[ρ1(V )WA − ρ1(W )VA]σVA

which is obviously vertical.In second place we prove that the bracket of a complete lift and a

vertical section is vertical. Indeed,

[ηC, V ] =∑A

VA[ηC, σVA] +∑A

ρ1(ηC)VA σV

A

=∑A

VA[η, σA]V +∑A

ρ1(ηC)VA σV

A

which is also vertical.Finally, if X, Y are projectable and projects to σ and η, respectively,

then X = σC + V and Y = ηC +W for some vertical sections V and W .Therefore

τ2 ◦ [X,Y ] = τ2 ◦ [σ, η]C + τ2 ◦ [σC,W ]+τ2 ◦ [V, σC]+ τ2 ◦ [V,W ] = [σ, η]◦τ,because the last three terms are vertical.

An immediate consequence of this proposition is the following:

• The set of vertical sections is a Lie subalgebra of Sec(LE).

https://www.researchgate.net/publication/256206368_Algebraic_constructions_in_the_category_of_lie_algebroids?el=1_x_8&enrichId=rgreq-aa2b7411a9c06603fecc66aeb537919e-XXX&enrichSource=Y292ZXJQYWdlOzIyNjgzMDUyNztBUzoyMjAyMzYwNTA2MzY4MDJAMTQyOTUxOTc3NTAwNw==


• The set of projectable sections Secpr(LE) is a Lie subalgebra ofSec(LE) and τ2 is a homomorphism of Lie algebras.• The set of vertical sections is an ideal of Secpr(LE).

Using the local description given in the last section, the structure func-tions of LE are given by the following formulas

ρ1(Xα) = ρiα∂

∂xiρ1(Vα) =

∂

∂yα

[Xα,Xβ] = Cγαβ Xγ [Xα,Vβ] = 0 [Vα,Vβ] = 0.

If {X α,Vα} denotes the dual base of {Xα,Vα} then the local expressionof the differential of a function F on LE is

dF = ρiα∂F

∂xiX α +

∂F

∂yαVα.

In particular, we have dxi = ρiαX α and dyα = Vα. The differential ofsections of (LE)∗ is determined by

dX α = −1

2Cαβγ X β ∧ X γ and dVα = 0.

5. The Liouville section and the vertical endomorphism

Besides the basic geometry of LE studied in the preceding sections,there are two canonical objects on LE whose definition and propertiesmimic the ones of its corresponding objects in the tangent bundle. Theseare the Liouville section and the vertical endomorphism.

The Liouville section ∆ is the section of τ1 whose value at the point ais the vertical lift to the point a of a itself, that is,

∆(a) = ξV (a, a) = (a, 0, aVa ).

As the Liouville vectorfield in a vector bundle, the Liouville sectionmeasures the homogeneity of functions and sections. This is an obviousconsequence of the fact that ρ1(∆) is the Liouville vectorfield on E. Wehave the following immediate properties for a function f on M and asection θ of E∗

d∆f = 0 and d∆θ = θ.

For the vertical and complete lift of a section of E we have the followingresult.

Proposition: If σ is a section of E then we have

[∆, σV ] = −σV and [∆, σC] = 0

16 EDUARDO MARTINEZ

Proof. The first bracket is vertical because it is the bracket of two verti-cals. Therefore we have to prove that the action of the bracket on linearfunctions is equal to the action of −σV . If θ is a section of E∗ then

ρ1([∆, σV ])θ = d∆dσV θ−dσV d∆θ = d∆〈 θ , σ 〉−dσV θ = −dσV θ = −ρ1(σV )θ.

This proves the first. For the second we note first that the bracket isvertical since it is the bracket of a projectable and a vertical. On linearfunctions we have

ρ1([∆, σC])θ = d∆dσC θ − dσCd∆θ = d∆dσθ − dσC θ = dσθ − dσθ = 0,

which proves the second relation.

The second important object is the vertical endomorphism S. It is theendomorphism of τ1 : LE → E defined by projection followed by verticallifting S = ξV ◦ τ12, or explicitely

S(a, b, v) = (a, 0, bVa ).

An immediate consequence of the definition is that

S(σV ) = 0 and S(σC) = σV

for any section σ of E.

Proposition: The vertical endomorphism satisfies S2 = 0. Moreover,ImS = KerS = Ver(LE)

Proof. If σ is a section of E then S2(σV ) = 0 because S(σV ) = 0, andS2(σC) = S(σV ) = 0. Therefore S2 = 0, from where it follows ImS ⊂KerS. If V =

∑vAσ

CA +

∑wBη

VB is an element in the kernel of S

then S(V ) = 0 = vAσVA, from where if follows that vAσA = 0, and

therefore V =∑wBη

VB is vertical. If we take W =

∑wBη

CB, then

S(W ) =∑wBη

VB = V , so that V is in the image of S.

Proposition: The vertical endomorphism is homogeneous of degree−1:

[∆, S] = −S.

Proof. On vertical sections

[∆, S](σV ) = [∆, S(σV )]− S([∆, σV ]) = −S(−σV ) = 0,

and on complete lifts

[∆, S](σC) = [∆, S(σC)]− S([∆, σC]) = [∆, σV ] = −σV

and therefore [∆, S] = −S.


We recall that the Nijenhuis tensor of an endomorphism A is definedby NA = 1

2[A,A], or explicitely

NA(X,Y ) = [A(X), A(Y )]− A([A(X), Y ])− A([X,A(Y )]) + A2([X,Y ]),

for X and Y sections of LE.

Proposition: The Nijenhuis tensor of the vertical endomorphism van-ishes.

Proof. Since S2 = 0, we have to prove that

NS(X,Y ) = [S(X), S(Y )]− S([S(X), Y ])− S([X,S(Y )]) = 0

for every pair of sections X, Y of LE. As usual it is enough to provethat relation for vertical and complete lifts. On two vertical lifts it clearlyvanishes since every one of the three terms vanishes. On a vertical anda complete lift we have

NS(σV , ηC) = −S([σV , ηV ] = 0

Finally, for two complete lifts

NS(σC, ηC) = [σV , ηV ]− S([σV , ηC])− S([σC, ηV ]) = −2S([σ, η]V ) = 0.

Therefore NS = 0.

The bracket of the vertical endomorphism with a vertical or a completelift vanishes

[σV , S] = 0 and [σC, S] = 0.

The proof proceeds as in the case E = TM and will be omitted (see [2]).

The coordinate expressions of ∆ and ρ1(∆) are

∆ = yαVα ρ1(∆) = yα∂

∂yα.

and the local expression of S is

S = Vα ⊗X α,

where {X α,Vα} is the dual base of {Xα,Vα}.


18 EDUARDO MARTINEZ

6. Second-order differential equations

In the case E = TM there are two equivalent definitions of a second-order differential equation on a manifold. The first one defines it asa vector field on TM such that their integral curves are the naturalprolongation of curves on the base manifold M . The second one statesthat it is a vector field Γ satisfying S(Γ) = ∆.

In the case of a general Lie algebroid the notion of admissible curvesreplaces that of natural prolongation.

Definition: A tangent vector v to E at a point a is called admissibleif Taτ(v) = ρ(a). A curve in E is admissible if its tangent vectors areadmissible. The set of all admissible tangent vectors will be denotedAdm(E).

Notice that v is admissible if and only if (a, a, v) is in LE. Thereforewe will consider Adm(E) as the subset of LE of all the elements of thatform, that is

Adm(E) = { z ∈ LE | τ1(z) = τ2(z) }

This definition mimics that of the second-order tangent bundle T 2Mto a manifold M as the diagonal of T (TM), that is, the set of vectorsv ∈ T (TM) such that τTM(v) = TτM(v). Therefore we consider Adm(E)as a substitute for T 2M .

Proposition: The following properties are equivalent for a section Γ ofLE

1. Γ takes values in Adm(E)2. τ2 ◦ Γ = idE3. S(Γ) = ∆.

A section of LE satisfying one of the above properties is called a second-order differential equation (sode) on the Lie algebroid E.

Proof. If Γ takes values in Adm(E) then τ2 ◦ Γ = τ1 ◦ Γ = idE, since Γ isa section of LE. Conversely, if τ2 ◦ Γ = idE, since also τ1 ◦ Γ = idE, wehave that τ2 ◦ Γ = τ1 ◦ Γ, which is the condition for the image of Γ to bein Adm(E). This proves the equivalence of the first two conditions.

We now prove the equivalence of the first and the third. If Γ(a) =(a, a, v) then S(Γ(a)) = (a, 0, aVa ) = ∆(a). Conversely, if Γ(a) = (a, b, v)then S(Γ(a)) = (a, 0, bVa ) and ∆(a) = (a, 0, aVa ), and the equality S(Γ) =∆ implies aVa = bVa . Since the vertical lift is an isomorphism we have thata = b, and therefore Γ(a) = (a, a, v) ∈ Adm(E).


It should be noticed that our definition differs slightly from that ofWeinstein [10]. He considers a sode as a special vector field on E, whilewe prefer to consider a sode as a special section of LE. The sode

vectorfield is obviously the image of the sode section by the anchor ρ1.Nevertheless it is important to note that if X is a section of LE suchthat ρ1(X) is a sode vector field then it is not true that X is a sode

section. For instance, if σ is a section of E which is in the kernel of ρ,and Γ is a sode section, then X = Γ + σC, is not a sode section whileρ1(X) is a sode vector field.

In local coordinates, a sode on E has the expression

Γ(x, y) = yαXα + fα(x, y)Vαand the associated vector field is of the form

ρ1(Γ)(x, y) = ρiαyα ∂

∂xi

∣∣∣(x,y)

+ fα(x, y)∂

∂yα

∣∣∣(x,y)

.

The integral curves of the sode Γ, i. e. the integral curves of ρ1(Γ),satisfy the differential equations

dxi

dt= ρiα(x)yα

dyα

dt= fα(x, y).

7. Lagrangian formalism

When a Lagrangian L ∈ C∞(E) is given on the Lie algebroid E, wecan define a dynamical system on E. This was done by Weinstein [10]in two different ways. If the Lagrangian is regular we can pull-back theHamiltonian system on E∗ by the Legendre transformation. Alterna-tively one can do variational calculus finding the extremals of the actionfunctional J =

∫ t1t0Ldt restricted to admissible curves.

The equations defining such dynamical system are the Euler-Lagrangeequations, which in local coordinates are

dxi

dt= ρiαy

α

d

dt

(∂L

∂yα

)= ρiα

∂L

∂xi− Cγ

αβyβ ∂L

∂yγ.

The purpose of this section is to put the Lagrangian formalism in ageometric framework, which allows to find the Euler-Lagrange equationsin a direct way, following the work of Klein [4], without any referenceto the structures in the dual. We will show that it is possible to definea symplectic structure (pre-symplectic, if the Lagrangian is singular) on

20 EDUARDO MARTINEZ

the bundle τ1 : LE → E by means of which we will find the dynamics bya symplectic equation. In particular this formalism will allow to studythe case of singular Lagrangians.

We will proceed by defining first the analog of the Cartan 1-form. Thenthe analog of the symplectic form (the Cartan 2-form) is the differential ofthe Cartan 1-form, and the Euler-Lagrange equations are defined in termsof the energy and the symplectic structure. Of course, the differential wemention is the differential in the Lie algebroid.

The analog of a differential 1-form in our framework is a section of thethe dual bundle (LE)∗. Therefore, we define the Cartan 1-section θL by

θL = S(dL).

The action on vertical and complete lifts is given by

〈 θL , σC 〉 = dσV L and 〈 θL , σV 〉 = 0.

It is clear that θL is a semibasic section, in the sense that it vanisheswhen restricted to Ver(LE). It follows that we can identify θL with amap from E to E∗, which is but the Legendre transformation. In localcoordinates

θL =∂L

∂yαX α.

The Cartan 2-section is (minus) the differential of θL

ωL = −dθLFrom the local expression of θL we find

ωL =∂2L

∂yα∂yβX α ∧Vβ +

1

2

(∂2L

∂xi∂yαρiβ −

∂2L

∂xi∂yβρiα +

∂L

∂yγCγαβ

)X α ∧X β

We will say that L is regular if ωL is regular as a bilinear form at everypoint. From the local expression of ωL it is clear that L is regular if and

only if the symmetric matrix gαβ =∂2L

∂yα∂yβis regular.

The energy function EL defined by the Lagrangian L is

EL = d∆L− L,which in local coordinates is

EL =∂L

∂yαyα − L

In terms of this objects we set the symplectic equation

iΓωL = dEL,


for a section Γ of LE. As we will readily show, the Euler-Lagrangeequations for L are the equations for the integral curves of Γ.

Indeed, if we put Γ = gαX α + fαVα then

iΓωL = gβ∂2L

∂yα∂yβVα−

[fβ

∂2L

∂yα∂yβ+ gβ

(∂2L

∂xi∂yαρiβ −

∂2L

∂xi∂yβρiα +

∂L

∂yγCγαβ

)]X α

and

dEL =∂2L

∂yα∂yβyβVα −

(ρiα∂L

∂xi− ρiα

∂2L

∂xi∂yβyβ)X α.

The equality of the Vα components implies

gαβ(yα − gα) = 0.

If the Lagrangian is regular, then this equation has a unique solutiongα = yα, which implies that Γ is sode. In the singular case, we have toimpose this as an additional condition, as it happens in the case of theusual Lagrangian Mechanics on TM . Taking this into account, the X α

components are equal if

ρiβyβ ∂2L

∂xiyα+ fβ

∂2L

∂yαyβ= ρiα

∂L

∂xi− Cγ

αβyβ ∂L

∂yγ.

In the left hand side of this equation we recognize the derivative of ∂L∂yα

along Γ, so that it can be written in the form

dΓ

(∂L

∂yα

)= ρiα

∂L

∂xi− Cγ

αβyβ ∂L

∂yγ,

which is the second of the Euler-Lagrange equations given by Weinstein.If Γ is a sode (or the Lagrangian is regular) then the Euler-Lagrange

equations can be expressed in an equivalent way as

dΓθL = dL,

because

iΓωL − dEL = dL− dΓθL.

The formalism given above is symplectic, while the Hamiltonian systemon E∗ defined by Weinstein is given in terms of the canonical Poissonstructure on E∗. This is an indication that it must be possible to definea canonical symplectic structure on a bundle over E∗ by means of which itis possible to express the Poisson bracket. This will be studied elsewhere.In the Lagrangian counterpart, the Poisson bracket of two functions Fand G on E is given now by the usual rule in terms of the symplectic

22 EDUARDO MARTINEZ

structure. We consider the Hamiltonian sections XF and XG of LEassociated to those functions

iXFωL = dF and iXGωL = dG.

Then we have that

ωL(XF , XG) = −{F,G},

as it can be easily checked in coordinates. The Jacobi identity for thePoisson bracket is, as usual, equivalent to the equation dωL = 0.

Alternatively, one can define a section δL of E∗ along τadm, called theEuler-Lagrange 1-section, as it was done in [1] for the case E = TM .For that we define the canonical section T as the identity map in E, andits prolongation T(1) as the inclusion of Adm(E) in LE, thought of assections along τ and τadm, respectively. In coordinates

T = yαeα and T(1) = yαXα + vαVα.

Moreover we define the differential operator dT(1) by

dT(1) = iT(1)d+ diT(1) .

In terms of this objects we have that f = iTdf = dTf and, more gen-erally, θ = iTθ, for a function f on M and a section θ of E∗. For afunction F on E we have that dT(1)F is the function on Adm(E) whichin coordinates is

dT(1)F = ρiαyα ∂F

∂xi+ vα

∂F

∂yα.

Then the Euler-Lagrange 1-section is

δL = dT(1)θL − dL

considered as a map from Adm(E) to E∗. In coordinates

δL =

[dT(1)

(∂L

∂yα

)− ρiα

∂L

∂xi+ Cγ

αβyβ ∂L

∂yγ

]eα.

A curve η is a solution of the Euler-Lagrange equations if it is admissibleand δL ◦ η = 0. In other words the set of points of Adm(E) in which δLvanishes is (if L is regular) a section of Adm(E), i. e. a sode.

https://www.researchgate.net/publication/230916265_A_new_approach_to_the_converse_of_Noether's_theorem?el=1_x_8&enrichId=rgreq-aa2b7411a9c06603fecc66aeb537919e-XXX&enrichSource=Y292ZXJQYWdlOzIyNjgzMDUyNztBUzoyMjAyMzYwNTA2MzY4MDJAMTQyOTUxOTc3NTAwNw==


8. Noether’s theorem

As an application, we will show how Noether’s theorem can be ex-tended to Lagrangian systems on Lie algebroids. As the original Noether’stheorem it associates a first integral to a symmetry of the action. SinceθL is semibasic, we will use the notation 〈 θL , σ 〉 to denote the function〈 θL , σC 〉. We will assume that the Lagrangian is regular.

Theorem: Let σ be a section of E and f a function on M such that

dσCL = f .

Then the function G = 〈 θL , σ 〉 − f is a first integral for the dynamics Γdefined by the Lagrangian L. Moreover, σC is a symmetry of Γ, that is[σC,Γ] = 0.

Proof. Let Γ be the second order differential equation defined by theLagrangian. Taking into account that [σC,Γ] is vertical we have

0 = 〈 dΓθL − dL , σC 〉 = dΓ〈 θL , σC 〉 − 〈 θL , [Γ, σC] 〉 − dσCL= dΓ〈 θL , σC 〉 − dσCL.

Therefore, if dσCL = f , we have that

dΓ〈 θL , σC 〉 − dΓf = 0,

where we have used that dΓf = f . Moreover, since [σC, S] = 0 we have

that dσCθL = df , and therefore iσCωL = dG. Hence,

i[Γ,σC ]ωL = dΓiσCωL − iσCdΓωL = dΓdG = ddΓG = 0,

and since L is regular, we have that [σC,Γ] = 0.

As it is well known in the case E = TM , Noether’s theorem in theversion given above does not establish a one to one correspondence be-tween first integrals and symmetries of the dynamical system. In orderto have a one to one relation we have to extend the notion of symmetryof the Lagrangian as in [1].

In order to do that we have to define the prolongation of a section Xof E along τ to a section X(1) of LE along τadm : Adm(E)→ E. This isdone by saying that X(1) projects to X

τ2 ◦X(1) = X ◦ τadm,

and that X(1) commutes with T(1) in the sense

dX(1)iT = iT(1)dX .


24 EDUARDO MARTINEZ

This conditions extends the construction of the complete lift of a sectionσ of E. Indeed, if X = σ ◦ τ then X(1) = σC ◦ τadm. In local coordinates,if X = Xα(x, y)eα then

X(1) = XαXα + (dT(1)Xα − CαβγX

βyγ)Vα.

Then it is easy to prove a global version of the variational equation

dX(1)L = −〈 δL ,X 〉+ dT(1)〈 θL , X 〉,

from where we have the following.

Theorem: If X is a section of E along τ and F is a function on E suchthat

dX(1)L = dT(1)F,

then G = 〈 θL , X 〉 − F is a constant of the motion and X(1) ◦ Γ is asymmetry for the dynamics defined by L.

Conversely, if G is a constant of the motion for the dynamics definedby the Lagrangian L and Y is the corresponding Hamiltonian section,iY ωL = dG, then

dX(1)L = dT(1)F,

where X = τ2 ◦ Y and F = 〈 θL , X 〉 −G.

The proof is a literal translation of the one given on [1]. We refer tothe reader to that paper for the details.

9. Examples

We consider a Lie algebra g acting on a manifold M , that is, we havea Lie algebra homomorphism g → X(M) mapping every element ξ ofg to a vectorfield ξM on M . The bundle E is E = M × g with theprojection onto the first factor. The anchor is the map ρ(m, ξ) = ξM(m).The bracket is defined by declaring ρ to be the anchor and defining thebracket of constant sections as the constant section corresponding tothe bracket on g, that is, if σ(m) = (m, ξ) and η(m) = (m, ζ) are twoconstant sections, then [σ, η](m) = (m, [ξ, ζ]g).

By identifying TE ≡ TM × Tg ≡ TM × g× g, an element of LE is ofthe form

(a, b, v) =((m, ξ), (m, η), (vm, ξ, ζ)

)and the condition Tτ(v) = ρ(b) implies that vm = ηM(m). Therefore, wecan identify LE with M × g× g× g with the projection τ1 onto the first



two factors

τ1(m, ξ, η, ζ) = (m, ξ)

τ2(m, ξ, η, ζ) = (m, η)

ρ1(m, ξ, η, ζ) = (ηM(m), ξ, ζ)

Given a base {eα} of g the base {Xα,Vα} of sections of LE is given by

Xα(m, ξ) = (m, ξ, eα, 0) and Vα(m, ξ) = (m, ξ, 0, eα).

If σ is a section of E, it is of the form σ(m) = (m,λ(m)) for a functionλ : M → g. Then, the vertical lift of σ is

σV (m, ξ) = (m, ξ, 0, λ(m))

and the complete lift of σ is

σC(m, ξ) = (m, ξ, λ(m), ξMλ(m) + [ξ, λ(m)]g),

where ξMλ = ξiM∂λ∂xi

is the differential of λ along ξM .We consider a Lagrangian of mechanical type

L(m, ξ) =1

2g(ξ, ξ)− V (m),

where g is an inner product on the Lie algebra g and V is a function onM . If a = (m, ξ) is an element of E and z1, z2 are two elements of LEover the point a,

z1 = (m, ξ, η1, ζ1) and z1 = (m, ξ, η1, ζ1),

then the Cartan sections are given by

θL(z1) = g(ξ, η1)

andωL(z1, z2) = g(η1, ζ2)− g(ζ1, η2) + g(ξ, [η1, η2]g).

The differential of the energy is

dEL(z2) = g(ξ, ζ2) + 〈 dV (m) , η2 〉.Therefore if Γ(m, ξ) = (m, ξ, ξ, ξ) then

(iΓωL − dEL)(z2) = −g(ξ, η2) + g(ξ, [ξ, η2]g)− 〈 dV (m) , η2 〉.

If we define ad†ξ by

g(ad†ξ η1, η2) = g(η1, adξ η2) = g(η1, [ξ, η2]g),

and the gradient of V by

g(gradV (m), η) = 〈 dV (m) , η 〉,

26 EDUARDO MARTINEZ

then(iΓωL − dEL)(z2) = −g(ξ − ad†ξ ξ + gradV (m), η2),

from where we get that the sode Γ is

Γ(m, ξ) = (m, ξ, ξ, ad†ξ ξ − gradV (m)).

The integral curves (m(t), ξ(t)) of Γ are the solution of the diferentialequations

m = ξ(t)M

(m)

ξ − ad†ξ ξ = − gradV (m).

On the left hand side of the second equation we can recognize the covari-ant derivative ∇ξξ on the Lie algebra g (the one comming from reductionof the Levi-Civita connection on the Lie group G) and therefore we canwrite the Euler-Lagrange equations in the form

m = ξ(t)M

(m)

∇ξξ = − gradV (m).

As a particular example we can consider the heavy top, where g = so(3)and M = S2. An element of M will be considered as an unit vector γin R

3 (representing the direction of the gravity), and an element of so(3)will also be considered as a vector ω in R

3 (representing the angularvelocity in body coordinates). The metric g is given by the inertia tensorof the top, g(ω1, ω2) = ω1 · Iω2, and the potential is V (γ) = mglγ · e,where e is the unit vector from the fixed point to the center of mass. Theanchor map is

ρ(γ, ω) = γ × ω ≡ (γ, γ × ω) ∈ TγS2,

and the bracket is given by

[ω1, ω2]so(3) = ω1 × ω2.

Thenad†ω1

ω2 = I−1(Iω2 × ω1)

and

gradV = I−1(∂V

∂γ× γ) = −MglI−1(γ × e).

Therefore the equations of motion are

γ = γ × ωω + I−1(ω × Iω) = MglI−1(γ × e),


or equivalently

γ + ω × γ = 0

Iω + ω × Iω = Mglγ × e,

which are the Euler-Arnold equations.

We now apply Noether’s theorem to find some symmetries. We con-sider rotations arround the gravity axis. The generator is the section

σ(γ) = (γ, γ).

whose complete lift is

σC(γ, ω) = (γ, ω, γ, 0).

Therefore

dσCL = ρ(γ) · ∂L∂γ

= 0,

since ρ(γ) = 0. Thus σ is a symmetry of the system and the constant ofmotion is

Jz = 〈 θL , σ 〉 =∂L

∂ω· γ = (Iω) · γ,

which is the component of the angular momenta in the direction of thegravity.

If the body is symmetric, that is Ie = I3e and I1 = I2, then the section

η(γ) = (γ, e)

is also a symmetry. Indeed, the complete lift of η is

ηC(γ, ω) = (γ, ω, e, ω × e),

and then

dσCL = (γ × e) · ∂L∂γ

+ (ω × e) · ∂L∂ω

= (ω × e) · Iω,

which vanishes by virtue of the symmetry of the body. The associatedfirst integral is

J3 = 〈 θL , η 〉 =∂L

∂ω· e = (Iω) · e.

It can be seen that σ is a symmetry for any Lagrangian system definedon S2 × so(3). This is due to the fact that σ is in the center of the Liealgebroid (that is, commutes with every other section). It is easy to seethat a section is in the center if and only if its complete lift is in thekernel of ρ1. Thus dσCL = ρ1(σC)L = 0 for any function L on E. Ofcourse, the constant of motion depends of the Lagrangian. Therefore, in

28 EDUARDO MARTINEZ

the category of Lie algbroids the center of the algebroid is the analog ofthe set of Casimir functions in the category of Poisson manifolds.

10. Conclusions and outlook

We have developed a geometric formalism for Lagrangian systems onLie algebroids. Our theory is formally identical to the usual one on thetangent bundle at the only cost of working with sections of LE and (LE)∗

instead of working with vectorfields and differential forms on a manifold.Therefore, nearly any result known to be true in the ordinary Lagrangianmechanics will also hold in this generalized framework. In particular ourformalism is, not only Poisson, but symplectic. This is an indication thatthe Hamiltonian counterpart can be developed in a parallel way. Indeedit is possible to define a prolongation of the dual bundle E∗ →M wherea canonical exact two form exists. The Poisson structure is then definedin terms of the symplectic one in the usual manner. This is the subjectof a forthcoming paper [8].

In the case of the canonical Lie algebroid E = TM , a second orderdifferential equation defines a nonlinear connection on TM →M (see [3])and, by a kind of linearization, a linear connection on TM ×M TM →TM , see [7]. Using this connection we found a differential operator,called the dynamical covariant derivative, and an endomorphism, calledthe Jacoby endomorphism, in terms of which the equation for Jacobifields is conveniently expressed. The generalization of this theory to thecase of a general Lie algebroids is under development.

Weinstein studies the problem of reduction for systems on Lie alge-broids. In the light of the symplectic nature of our theory, it is naturalto study how the symplectic form reduces and how this is related tosymplectic and Poisson reduction.Akcnowledgements: I would like to acknowledge to Frans Cantrijn,Jose Carinena, Janusz Graboswski, Carlos Lopez and Willy Sarlet forvery helpful discussions. Partial financial support from CICYT is ac-knowledged.

References

[1] Carinena JF, Lopez C and Martinez E, A new approach to the converse ofNoether’s theorem, J. Phys. A: Math. Gen. 22 (1989) 4777–4786.

[2] Crampin M, Tangent bundle geometry for Lagrangian dynamics, J. Phys. A:Math. Gen. 16 (1983) 3755–3772.

[3] Grifone J, Structure presque tangente et connections, Ann. Inst. Fourier 22(1) (1972) 287–334.

https://www.researchgate.net/publication/231980919_Geometric_characterization_of_linearisable_second-order_differential_equations?el=1_x_8&enrichId=rgreq-aa2b7411a9c06603fecc66aeb537919e-XXX&enrichSource=Y292ZXJQYWdlOzIyNjgzMDUyNztBUzoyMjAyMzYwNTA2MzY4MDJAMTQyOTUxOTc3NTAwNw==





https://www.researchgate.net/publication/247661474_Structure_presque_tangente_et_connexions_II?el=1_x_8&enrichId=rgreq-aa2b7411a9c06603fecc66aeb537919e-XXX&enrichSource=Y292ZXJQYWdlOzIyNjgzMDUyNztBUzoyMjAyMzYwNTA2MzY4MDJAMTQyOTUxOTc3NTAwNw==




[4] Klein J, Espaces variationnels et mecanique, Ann. Inst. Fourier 12 (1962) 1–124.[5] Liberman P, Lie algebroids and Mechanics, Arch. Math. (Brno) 32 (1996) 147–

162.[6] Higgins PJ and Mackenzie K, Algebraic constructions in the category of Lie

algebroids, J. of Algebra 129 (1990) 194–230.[7] Martinez E and Carinena JF, Geometric characterization of linearizable

second-order differential equations, Math. Procs. Camb. Phil. Soc. 119 (1996)373–381.

[8] Martinez E, Hamiltonian Mechanics on Lie Algebroids, (preprint).[9] Nijenhuis A, Vector forms brackets in Lie algebroids, Arch. Math. (Brno) 32

(1996) 317–323.[10] Weinstein A, Lagrangian Mechanics and groupoids, Fields Inst. Comm. 7

(1996) 207–231.

Eduardo Martınez

Departamento de Matematica Aplicada

Centro Politecnico Superior de Ingenierıa

Universidad de Zaragoza

Marıa de Luna 3, 50015 Zaragoza, Spain

E-mail address: [email protected]








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Lagrangian Mechanics on Lie Algebroids