The Hamiltonian Dynamical System Associated to the Isotropic … · 2014-01-22 · gineering. We use appropriate numerical methods for the study of conservation laws of Hamiltonian

The Hamiltonian Dynamical System Associated to the IsotropicHarmonic Oscillator - A Numerical Study of Conservation Laws

Romulus Militaru, Florian Munteanu

Abstract—-In this paper we study the interplay be-tween dynamical systems geometrical theory and compu-tational calculus of dynamical systems. The viewpoint isgeometric and we also compute and characterize objects ofdynamical significance, in order to understanding the math-ematical properties observed in numerical computation fordynamical models arising in mathematics, science and en-gineering. We use appropriate numerical methods for thestudy of conservation laws of Hamiltonian dynamical sys-tem associated to the isotropic harmonic oscillator for oneand two dimensional case.

Keywords—-Conservation law, Noether theorem,Runge-Kutta methods, symmetry.

I. INTRODUCTION

Symmetries play a key role in mathematics, physicsand very specially in mechanics. The study of those quan-tities which are conserved by a mechanical system is highlyrelevant both from a theoretical and from a practical pointof view ([8], [28]). Different methods for finding suchconserved quantities are known, those based on Noethertheory ([26]), which addresses the invariance of the ac-tion functional under infinitesimal transformations, haveproved powerful and widely used. For theoretical geomet-rical models, numerical analysis of models and more com-putation details see also [4], [5], [12], [25], [21], [22].

The inverse problem of Lagrangian mechanics is thequestion ([1], [30]): given a system of second-order ordi-nary differential equations, under what circumstances doesthere exist a regular Lagrangian function, such that the cor-responding Lagrange equations are equivalent (i.e. havethe same solutions) as the original equations. Locally, thequestion can be translated immediately into more preciseterms as follows: considering a given second-order systemin normal form qi = f i(q, q), which (for the time being)

This work was supported in part by EU Grant FP7-PEOPLE-2012-IRSES-316338 and by the Faculty of Electrical Engineering of Universityof Craiova, Romania.

R. Militaru is with Department of Applied Mathematics, Universityof Craiova, Al. I. Cuza st., no. 13, 200585-Craiova, Romania (corre-sponding author to provide phone: +40726003426; e-mail: [email protected]).

F. Munteanu is with Department of Applied Mathematics, Uni-

versity of Craiova, Al. I. Cuza 13, 200585-Craiova, Romania

([email protected]).

we take to be autonomous for simplicity, what are theconditions for the existence of a symmetric, non-singularmultiplier matrix (gij(q, q)) such that

gij(qj − f j(q, q)

)=

d

dt

(∂L

∂qi

)− ∂L

∂qi

for some L. Clearly (gij), if it exists, will become the Hes-sian of the Lagrangian L. The literature on this problem isextensive ([8]); the conditions for the existence of L areusually referred to as the Helmholtz conditions, but thesecan take many different forms depending on the mathemati-cal tools one uses and on the feature one focusses on ([27]).

In this paper we will study symmetries, conservationlaws and relationship between this in the geometric frame-work of Classical Mechanics ([1], [2], [8], [23], [24]).Also, we will use appropriate numerical methods for thestudy of the behavior of dynamical system associated to theisotropic harmonic oscillator for one dimensional case andtwo dimensional case. For theoretical geometrical models,numerical analysis of models and more computation detailssee also [4], [5], [12], [21], [22].

There is a very well-known way to obtain conserva-tion laws for a system of differential equations given by avariational principle: the use of the Noether Theorem ([26])which associates to every symmetry a conservation law andconversely. However, there is a method introduced by G.L.Jones ([19]) and M. Crasmareanu ([10]) which can be ob-tained new kinds of conservation laws, without the help ofa Noether’s type theorem.

In the second section we recall the basic notions andresults for the geometrical study of a Hamiltonian dynam-ical systems, we present the classical Noether Theorem([26]), the Theorem of Jones-Crasmareanu ([19], [10]) and,finally, we focus on two examples: 1D and 2D isotropicharmonic oscillator ([10], [30]).

In the third section, we will make a numerical studyfor the isotropic harmonic oscillator. More precisely, con-structing a Matlab-based numerical code, we are looking toapproximate and characterize different types of invariantsand also to extract statistical information on the dynami-cal behavior and perform some comparisons for differentinitial conditions associated to the considered problem.

All manifolds are real, paracompact, connected andC∞. All maps are C∞. Sum over crossed repeated indicesis understood.

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II. CLASSICAL RESULTS

A. Symmetries and Conservation Laws for Dynam-ical Systems Let M be a smooth, n-dimensional mani-

fold, C∞(M) the ring of real-valued smooth functions,X (M) the Lie algebra of vector fields and Ap(M) theC∞(M)-module of p-differential forms, 1 ≤ p ≤ n. ForX ∈ X (M) with local expression X = Xi(x) ∂

∂xi we con-sider the system of ordinary differential equations whichgive the flow {Φt}t of X , locally,

xi(t) =dxi

dt(t) = Xi(x1(t), . . . , xn(t)) . (1)

A dynamical system is a couple (M,X), where M is asmooth manifold and X ∈ X (M). A dynamical systemis denoted by the flow of X , {Φt}t or by the system ofdifferential equations (1).

A function f ∈ C∞(M) is called conservation law fordynamical system (M,X) if f is constant along the everyintegral curves of X , that is

LXf = 0, (2)

where LXf means the Lie derivative of f with respect toX .A diffeomorfism Φ : M → M is said to be a symmetry of(M,X) if Φ maps integral curves of X onto integral curvesof X , i.e., TΦ(X) = X .Y ∈ X (M) is called dynamical symmetry (or, shortly, sym-metry) for (M,X) if its flow {Φt}t consists of symmetriesof (M,X), or, equivalently, [Y,X] = LXY = 0.

Let us recall that ω ∈ Ap(M) is called invariant formfor (M,X) if LXω = 0.

B. Hamiltonian Systems If (M,ω) is a symplectic

manifold then the dynamical system (M,X) is said to bea dynamical Hamiltonian system (or, shortly, Hamiltoniansystem) if there exists a function H ∈ C∞(M) (called theHamiltonian) such that

iXω = −dH, (3)

where iX denotes the interior product with respect to X .The vector field X from (3) is called the Hamiltonian vec-tor field associated to H (also denoted by XH ) and

XH =∂H

∂pi

∂

∂xi− ∂H

∂xi

∂

∂pi

with respect to natural local coordinates (xi, pi), given byDarboux Theorem, ([23],[8]).The equation (3) is equivalent, locally, with Hamilton-Jacobi equations:

dxi

dt=

∂H

∂pi,dpidt

= −∂H

∂xi.

A regular Lagrangian L on TM is a smooth function L :

TM → R, for which the fundamental metric tensor gij =∂2L

∂yi∂yj has rank n on TM \ {0}.Let us remember the Legendre map associated to L, FL :TM → T ∗M , which is a local diffeomorphism given by(xi, yi) −→ (xi, pi), where pi =

∂L∂yi .

Let us consider the Hamiltonian system (TM,SL) onthe symplectic manifold (TM,ωL), where SL is the canon-ical semispray associated to the regular Lagrangian L, lo-cally, given by

SL = yi∂

∂xi−Gi ∂

∂xi,

where Gi = gij(

∂2L∂yj∂xk y

k − ∂L∂xj

)and gij being the in-

verse of the fundamental metric of L, gij , [23].In fact, the Euler-Lagrange equations associated to L,

Ei(L) =d

dt

(∂L

∂yi

)− ∂L

∂xi= 0

are equivalently with the system of ODE’s,

dyi

dt= −Gi,

dxi

dt= yi ,

which give the flow of vector field SL on TM .ωL = dθL is the Cartan 2-form of L, θL = J∗(dL), J∗

being the adjoint of the natural tangent structure J on TMand

H = EL =∂L

∂yiyi − L

is the energy of L or the Hamiltonian of (TM,SL).A Cartan symmetry for Lagrangian L is a vector field

X ∈ X (TM) characterized by LXωL = 0 and LXH = 0.It is known that that any Cartan symmetry for LagrangianL is a symmetry for the canonical semispray S of L.

For each Cartan symmetry X for (M,L) we havedLXθL = 0, which implies that LXθL is a closed 1-form.If LXθL is a exact 1-form, then we say that X is exact Car-tan symmetry for (M,L).Obviously, the canonical semispray of L is an exact Cartansymmetry for Lagrangian L, [8],[23], [28].

C. Noether Theorem

In the classical case (k = 1), we know that Cartansymmetries induce and are induced by constants of motions(conservation laws), and these results are known as NoetherTheorem and its converse.

Theorem 1 ([26]) If X is an exact Cartan symmetry withLXθL = df , then

PX = J(X)L− f

is a conservation law for the Euler-Lagrange equations as-sociated to the regular Lagrangian L.


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Conversely, if F is a conservation law for the Euler-Lagrange equations associated to the regular LagrangianL, then the vector field X uniquely defined by

iXωL = −dF

is an exact Cartan symmetry.

D. Nonclassical Conservation Laws

If Z ∈ X (M) is fixed, then Y ∈ X (M) is called Z-pseudosymmetry for (M,X) if there exists f ∈ C∞(M)such that LXY = fZ.A X-pseudosymmetry for X is called pseudosymmetry for(M,X).

Example: ([11],[13]) The Nahm’s system from thetheory of static SU(2)-monopoles:

dx1

dt= x2x3,

dx2

dt= x3x1,

dx3

dt= x1x2. (4)

The vector field X = x2x3 ∂∂x1 + x3x1 ∂

∂x2 + x1x2 ∂∂x3

satisfies [Y,X] = X , where Y =3∑

i=1

xi ∂∂xi .

This means that Y is a X-pseudosymmetry for (4).The next theorem gives the association between pseu-

dosymmetries and conservation laws. In this way, we willobtain new kinds of conservation laws, nonclassical, with-out the help of Noether’s type theorem.

Theorem 2 ([10],[19]) Let X ∈ X (M) be a fixed vectorfield and ω ∈ Ap(M) be a invariant p-form for X . If Y ∈X (M) is symmetry for X and S1, . . ., Sp−1 ∈ X (M) are(p− 1) Y -pseudosymmetry for X then

Φ = ω(X,S1, . . . , Sp−1) (5)

or, locally,

Φ = Si11 · · ·Sip−1

p−1 Yipωi1...ip−1ip (6)

is a conservation laws for (M,X).Particularly, if Y , S1, . . ., Sp−1 are symmetries for X

then Φ given by (5) is conservation laws for (M,X).

Now, we can apply this result to the dynamical Hamiltoniansystems.

Proposition 3 Let be (M,XH) a Hamiltonian system onthe symplectic manifold (M,ω), with the local coordinates(xi, pi). If Y ∈ X (M) is a symmetry for XH and Z ∈X (M) is a Y -pseudosymmetry for XH then

Φ = ω(Y, Z) (7)

is a conservation law for the Hamiltonian system(M,XH).

Particularly, if Y and Z are symmetries for XH thenΦ from ( 7) is a conservation law for (M,XH).

If Y = Y k ∂∂xk + Yk

∂∂pk

and Z = Zk ∂∂xk + Zk

∂∂pk

then(7) becomes

Φ =(

Y k Yk

)(0 −11 0

)(Zk

Zk

)=

YkZk−

−Y kZk.(8)

Corollary 4 If Y ∈ X (M) is a XH -pseudosymmetry forXH then

Φ = ω(XH , Y ) = −LY H (9)

or

Φ =∂H

∂xkY k +

∂H

∂pkYk (10)

is a conservation law for (M,XH).

E. Nonclassical Conservation Laws for Regular La-grangian Systems

If we consider the Hamiltonian system (TM,SL) onthe symplectic manifold (TM,ωL), where SL is the canon-ical semispray and ωL the Cartan 2-form associated to aregular Lagrangian L on TM , then we have:

Corollary 5 If Y = Y k ∂∂xk + Y k ∂

∂yk ∈ X (TM) is a SL-pseudosymmetry for SL then

Φ = ωL(SL, Y ) = −LY EL (11)

or

Φ =∂EL

∂xkY k +

∂EL

∂ykY k (12)

is a conservation law for (TM,SL).

An immediately consequence of this last result is the fol-lowing:

Corollary 6 If the canonical semispray SL associated tothe regular Lagrangian L is 2-positive homogeneous withrespect to velocity (SL is a spray) and gij is the metrictensor of L, then Φ = gijy

iY j is a conservation law for(TM,SL).

The canonical semispray SL is a spray if and only if[SL, C] = SL, that is LSLC = SL. So, in this case, thecanonical vector field C = yi ∂

∂yi is a pseudosymmetry forSL and using the last corollary we obtain the conservationof the kinetic energy T = 1

2gijyiyj . This is the case when

L = F 2, where F is a Finsler function like in [23].

F. Examples

Example: 1D harmonic oscillatorThe dynamics of the 1-dimensional harmonic oscilla-

tor is given by second-order ordinary differential equation

q + ω2 = 0 , (13)


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where ω2 = km , m is the mass of the oscillator (usualy we

take a standard, m = 1) and k is the elasticity constant.The period of the oscillations is T = 2π

ω .The Lagrangian is L(q, q) = T −V = 1

2

(q2 − ω2q2

),

where T is the kinetic energy T = 12 q

2 and V is the po-tential of the elastic force, V = 1

2kq2. The total energy

of L, EL = q ∂L∂q = 1

2

(q2 + ω2q2

)is a conservation law

for (13) and represents the Hamiltonian function of thedynamical system (13). The canonical semispray of L isSL = q ∂

∂q − ω2q ∂∂q .

If q(t) = A cosωt + Bω sinωt is the solution of

the Euler-Lagrange equation (13) with initial conditionsq(0) = A, q(0) = B, then we have

L = − 12

(A2ω2 +B2

)cos 2ωt−AB sin 2ωt ,

H = 12

(A2ω2 +B2

),

T = 12

(A2ω2 sin2 ωt+B2 cos2 ωt

).

Example: 2D isotropic harmonic oscillator

Let the 2-dimensional isotropic harmonic oscillator

q1 + ω2q1 = 0q2 + ω2q2 = 0

(14)

a toy model for many methods to finding conservation laws.The Lagrangian is

L =1

2

[(q1)2

+(q2)2]− ω2

2

[(q1)2

+(q2)2]

Next, by applying the conservation of energy we have twoconservation laws Φ1 =

(q1)2

+ ω2(q1)2, Φ2 =

(q2)2

+

ω2(q2)2

. Let us observe that 12 (Φ1 +Φ2) = H , the total

energy of the system.A straightforward computation give that the complet

lift of X = q2 ∂∂q1 − q1 ∂

∂q2 is an exact Cartan symmetrywith f = 0 and then the associated classical Noetherianconservation law is

Φ3 = PX = J(X)L = Xi ∂L

∂qi= q2q1 − q1q2.

Taking into account that the canonical spray of L is

S = q1∂

∂q1+ q2

∂

∂q2− ω2q1

∂

∂q1− ω2q2

∂

∂q2

and

Y = q2∂

∂q1+ q1

∂

∂q2− ω2q2

∂

∂q1− ω2q1

∂

∂q2

is a symmetry for S, it result that

Z = q1∂

∂q1+ q2

∂

∂q2+ q1

∂

∂q1+ q2

∂

∂q2

is a symmetry for S.Next, we have LY H = 0, LZH = 2H and then Φ =

ωL(S, Y ) = 0, Φ = ωL(S,Z) = 2H , that means that wenot have new conservation law applying main Theorem 2.But Φ4 = ωL(Y, Z) = q1q2 + ω2q1q2 is a new conserva-tion law given by the main Theorem 2 or by their corollar-ies.

We remark that Φ4 is a nonclassical conservation law,obtained by symmetries, and Φ4 represent the energy of anew Lagrangian of, L = q1q2 − ω2q1q2 ([30]).

Like in the 1D case, qi(t) = Ai cosωt +Bi

ω sinωt,i = 1, 2 is the solutions of the Euler-Lagrange equations(14) with initial conditions qi(0) = Ai, qi(0) = Bi, i =1, 2, and we have

L = −12

((A2

1 +A22

)ω2 +B2

1 +B22

)cos 2ωt−

− (A1B1 +A2B2) sin 2ωtH = qi ∂L

∂qi − L = 12

((A2

1 +A22

)ω2 +B2

1 +B22

)T = 1

2

((A2

1 +A22

)ω2 sin2 ωt+

(B2

1 +B22

)cos2 ωt

)Φ1 = A2

1ω2 +B2

1 ;Φ2 = A2

2ω2 +B2

2

H = 12 (Φ1 +Φ2) ;

Φ3 = 0Φ4 = HL = ω2A1A2 +B1B2

L = (−ω2A1A2 +B1B2) cos 2ωt−−ω(A1B2 +A2B1) sin 2ωt

Let us remark that Φ3 is the module of kinetic momentum,L = r × p.

III. NUMERICAL STUDY

In this section, constructing a Matlab-based numericalcode, we are looking to approximate and characterize dif-ferent types of invariants and also to extract informationson the dynamical behavior and perform comparisons forboth different initial conditions associated to the consid-ered problem and for different values of the parameter ω.In the first stage we focus on the numerical solving ofthe initial value problem given by the ordinary differentialequation with a prescribed initial condition, by appropriatenumerical methods, such as Runge-Kutta methods. Numer-ical integration is a mature subject, but due to the todayshigh computer efficiency it is still very active especiallywith regard to algorithms designed for special classes ofequations, [17], [29], [18]. Thus we obtain the numericalsolution represented by the approximate values of the exactsolution for a discrete set of data points.In the second stage, using this approach we perform a nu-merical analysis of the conservation laws and main sizes,like the Lagrangian, the Hamiltonian, the kinetic energy,the kinetic momentum.

A. One Dimensional Harmonic Oscillator

We consider the initial value problem (IVP) given by: q + ω2q = 0q(0) = 1q(0) = 0


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(a). ω = 0.59

Fig 1: Numerical solution on a period

Fig 2: Lagrangian L, function of t and q

Fig 3: Lagrangian L, as function of q, respectively functionof t

Fig 4: Total energy H , function of t and q

Fig 5: Total energy H , as function of q, respectively asfunction of t

Fig 6: Kinetic energy T , function of t and q

Fig 7: Kinetic energy T , as func-tion of q, respectively as function of t

(b). ω = 7.8


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Fig 8: Numerical solution on a period

Fig 9: Lagrangian L, function of t and q

Fig 10: Lagrangian L, as function of q, respectively asfunction of t

Fig 11: Total energy H , function of t and q

Fig 12: Total energy H , as function of q, respectively asfunction of t

Fig 13: Kinetic energy T , function of t and q

Fig 14: Kinetic energy T , as function of q, respectively asfunction of t


Issue 12, Volume 7, 2013 998

We present a comparison concerning the kinetic energyT for three different values of the parameter ω :

Fig 15: Profiles of the kinetic energy T, for three differentvalues of ω

Next we give the same comparison concerning the com-portment of the kinetic energy, but for another initial valueproblems obtained from the considered second order differ-ential equation characterizing the 1D harmonic oscillatorand different initial conditions.More precisely considering the following IVP: q + ω2q = 0

q(0) = 1q(0) = 1

the graphic profiles of kinetic energy are:


and also for the following IVP: q + ω2q = 0q(0) = 1q(0) = −1

the kinetic energy presents the graphic profiles ploted in thenext figure:


B. Two Dimensional Harmonic Oscillator

We consider the initial value problem given by :

q1 + ω2q1 = 0q2 + ω2q2 = 0q1(0) = 1q1(0) = 0q2(0) = 1q2(0) = 0

(a). ω = 0.59

Fig 18: Profile of the numerical solution q1 on a period



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Fig 20: Lagrangian L, function of t

Fig 21: Lagrangian L, function of q1, q2

Fig 22: Total energy H, function of q1, q2

Fig 23: Kinetic energy T, function of q1, q2

Fig 24: First conservation law Φ1, function of q1, q2

Fig 25: Second conservation law Φ2, function of q1, q2


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Fig 26: Third conservation law Φ3, function of q1, q2

Fig 27: Fourth conservation law Φ4, function of q1, q2

Fig 28: Lagrangian L function of q1, q2

(b). ω = 7.8



Fig 31: Lagrangian L, function of t


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Fig 32: Lagrangian L, function of q1, q2

Fig 33: Total energy H, function of q1, q2

Fig 34: Kinetic energy T, function of q1, q2

Fig 35: First conservation law Φ1, function of q1, q2

Fig 36: Second conservation law Φ2, function of q1, q2

Fig 37: Third conservation law Φ3, function of q1, q2


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Fig 38: Fourth conservation law Φ4, function of q1, q2

Fig 39: Lagrangian L function of q1, q2

Observation 7 The range of the values of the LagrangianL is symmetric around zero and the extreme values are di-rectly proportional with the values of parameter ω. Thiscan also be observed for the new Lagrangian L.

The extreme values of the kinetic energy T are also directlyproportional with the values of parameter ω. The depen-dance of the kinetic energy upon the initial conditions, forthe case of 1D harmonic oscillator, can be observed inFig.16 and Fig.17, while the influence of the values of theparameter ω on the comportment of the kinetic energy canbe analysed in Fig.15.The total energy H takes constant values for different val-ues of ω, as we presented in Fig.5, Fig.12, Fig.22 andFig.33. Thus we are in agreement with the property to be aconservation law.The others sizes as Φ1, Φ2, Φ3,Φ4, which are also lawsof conservation, are characterized through our numericalstudy, by constant values, depending on the values of ω,Fig.24-27 and Fig.35-38.

IV. CONCLUSION

We perform a numerical study of this mathematicalmodels, in order to approximate different types of invari-ants and main sizes, through numerical codes based onappropriate numerical calculus techniques. Thus, start-ing from certain initial value problems associated to ourmodels, we obtain the numerical solution and we developthe numerical characterization of the main sizes previouslyanalysed from the geometrical point of view. Thus we areable to make different comparisons between these studiedquantities for different values of parameters, for differentinitial conditions etc.

ACKNOWLEDGEMENTS

Both authors wish to thank the management of the Fac-ulty of Electrical Engineering, University of Craiova.

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The Hamiltonian Dynamical System Associated to the Isotropic … · 2014-01-22 · gineering. We use appropriate numerical methods for the study of conservation laws of Hamiltonian