Journal of Nonlinear Mathematical Physics

In Press, Corrected Proof, Available Online: 12 December 2020

Orbits and Lagrangian Symmetries on the Phase Space

Authors
Javier Pérez Álvare*
Departamento de Matemáticas Fundamentales, UNED, C/ Senda del Rey 9, 28040 Madrid, Spain
Corresponding Author
Javier Pérez Álvare
Received 24 August 2020, Accepted 26 November 2020, Available Online 12 December 2020.
DOI
https://doi.org/10.2991/jnmp.k.201203.001How to use a DOI?
Keywords
Lagrangian system, symmetry, conservation law
Abstract

In this article, given a regular Lagrangian system L on the phase space TM of the configuration manifold M and a 1-parameter group G of transformations of M whose lifting to TM preserve the canonical symplectic dynamics associated to L, we determine conditions so that its infinitesimal generator produces conservation laws, in terms of the orbits of G in TM.

Copyright
© 2020 The Author. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Variational geometry is a central theme in mathematics and in mathematical physics. In its development, it uses methods of the calculus of variations, differential geometry and global analysis and is based on different geometric structures: tangent spaces, fibered manifolds, etc. See for example the great work [13] for an intrinsic characterization of variational problems (variation formulas, symmetries and conserved quantities, Dirac theory of constraints, etc.) on fibered manifolds.

In the time independent case, Lagrangian theory, is a theory on the tangent bundle of the configuration space. Therefore, it is the geometry itself of the tangent bundle that offers its richness in the construction of a Lagrangian system. Thus, it is a well known fact that to every Lagrangian system on the phase space L:TM𝕉 it is possible to associate an exact 2-form wL on which Poisson algebra of functions on TM it is established the essential dynamical quantities of the system.

When the function L defines a regular Lagrangian system, the metric wL is irreducible, establishing in this way the base for a dynamical theory. We believe that this splendid construction of autonomous Variational Calculus starts with J. Klein [11, 12] and Grifone [9], providing a fundamental structure capable to place the theory in a unifying position in many local and global concepts of the Calculus of Variations (physical quantities, Noether invariants, infinitesimal symmetries, dynamical variables, etc.); it even allows to see its reflection in the multiphase construction of the Classical Field Theories (see, by example, [10]).

It is a remarkable fact that in this framework, the group of Noether symmetries of the Lagrangian system L, beyond being a group of point transformations it claims a deep geometrical meaning in terms of symplectic symmetries of the Hamiltonian dynamical system iζLwL=dEL.

It is not an easy task to find new invariants on the critical locus of a Variational problem. For Lagrangian systems some significant advances are Prince [15] Crampin [4], Marmo and Mukunda [14], de León and Martín [5]. Here, roughly speaking, we prove that for a 1-parameter group G of transformations of M whose action in (TM, wL) is symplectic, the corresponding infinitesimal generator produce first integrals of the motion for the ζL-dynamics whenever the Legendre function of the system is constant on the orbits of G.

2. SYMPLECTIC DYNAMICS OF LAGRANGIAN SYSTEMS

Let M be an n -dimensional differentiable manifold and TM its tangent bundle. Let us denote by πM: TMM the canonical projection. Given a coordinate neighborhood UM with local coordinates (xi), let us denote with TU the corresponding neighborhood in TM coordinated by (xi, vj). Given xM, and yTxM, a vector vTy(TN) such that πM*(v)=0 is called a vertical vector. A vector field X on TN is called vertical if Xy is a vertical vector for each yTN. An element uTxM determines a vertical vector at any point y in the πM -fiber over x, as the tangent vector at t = 0 to the curve ty + tu. Naturally, we can define the vertical lift Xv of a vector field X on N. Locally, if X = Xi∂/∂xi, then

Xv=Xi/vi.

On the other hand, if ϕt is a local 1-parameter group of transformations of M with the vector field X as its generator, then it has a natural lift as a local 1-parameter group t of transformations of TM whose generator is called the complete lift of X to TN and is denoted by Xc. In local coordinates, if X = Xi∂/∂xi then

Xc=Xi/xi+vjXi/xj/vi
(one may consult Crampin [4] or de Leon-Rodrigues [6]).

Let us consider the canonical almost tangent structure J on TM locally defined by the (1, 1) tensor field

J=vidxi.

We define the Liouville vector field C on TM as the one given in local coordinates by the expression

C=vivi.

Let us consider a regular Lagrangian function L:TM𝕉 , that is, the Hessian matrix in every coordinate neighborhood (xi, vj)

(2Lvivi),
is invertible. As a consequence, the 2-form on TM
wL=-dθL
where θL = dLJ is nondegenerate (see, for example, [7]).

In this way, we define the symplectic dynamics associated to the Lagrangian system L as the determined by the Hamilton equation

iζLwL=dEL, (2.1)
where EL = CLL is the energy function corresponding to L.

It is useful to consider the expression in local coordinates of the Legendre 1-form θL,

θL=iLvidxi.

The key fact is that the Hamiltonian vector field in (1) has the local coordinate expression

ζL=vixi+ak(xi,vj)vk (2.2)
where ak are smooth functions (1 ≤ kn). Consult Crampin [3] or Gotay and Nester [8]; also the monograph [6] goes through all the theory in its Chapter 7. This fact defines ζL as a second-order differential equation field; the projections of its integral curves onto M are the solutions of the system
xk=ak(xi,xj)
which comprise the Euler-Lagrange equations for the extremals of the Lagrangian L.

Finally, it easily follows from (2.2) the following expression for the Legendre function θL(ζL) of the Lagrangian system L,

θL(ζL)=CL. (2.3)

3. G-ORBITS vs. ζL-ORBITS: CONSERVATIONS LAWS

The search of constants of motion corresponding to 1-parameter groups of symmetries in classical mechanics, has had since Emmy Noether, a fruitful history. We turn to the problem of symmetry groups in mechanical Lagrangian systems by considering groups of evolution of the Hamiltonian dynamical system iζLwL=dEL. One may consult [6] for a modern version of the classical Noether theorem in our very context.

First of all we consider the Poisson structure associated to the symplectic phase space manifold (TM, wL). For every fC(TM), its Hamiltonian vector field is the unique vector field Xf on TM such that iXfwL=df. In this way, the Poisson bracket on C(TM) is given by

{f,g}=Xfg=-Xgf=-w(Xf,Xg),f,gC(TM).

It satisfies Leibniz’s rule

{f,gh}={f,g}h+g{f,h}
and the Jacobi’s identity
{f,{g,h}}+{g,{h,f}}+{h,{f,g}}=0.

We say that fC(TM) is a conservation law for the symplectic dynamics iζLwL=dEL if it belongs to the centralizer of EL for the Poisson bracket. We denote by S the set of all conservation laws

S={fC(TM):{f,EL}=0}.

It is clear, by the Jacobi’s identity, that S is stable under Poisson bracket.

On account of this structure, we can settle this work on the essential consideration of 1-parameter groups of transformations which permute the classical trajectories of the Lagrangian system defined by L:TM𝕉 . These symmetries, denoted dynamical symmetries by Prince [15] and Crampin [4], are crystallized by vector fields X𝔛(M) that act as infinitesimal symmetries of the second order differential equation field ζL,

[Xc,ζL]=0. (3.1)

Under these assumptions, Prince and Crampin (one may also consult the beautiful presentation on the subject in de Leon-Rodrigues [6]) with an effective combination of the vertical lift Xv and the complete lift Xc of the vector field X on M, raise a theory of conserved quantities of a Lagrangian system, under the condition that LXcθL is an exact 1-form df on TM and that XcEL = 0. Let us take as an example the conservation law

{f-XvL,EL}=0.

Our goal here is to find other conditions on the 1-parameter invariance groups, that submitted to the assumption (3.1) are able to generate first integrals on the trajectories of the Lagrangian system.

A natural condition, that comes from considering those transformations by diffeomorphisms of the Lagrangian L that lead to the same symplectic dynamics (see, for example, [1] or [2]), is

LXcθL=0. (3.2)

In such a case, we say that X𝔛(M) conforms a vector field on TM of Crampin-Prince symplectic symmetries.

As a direct consequence of this condition, we have

iXcwL=-iXcdθL=dθL(Xc). (3.3)

Theorem 3.1.

Let us consider a regular Lagrangian system L:TM𝕉 and X a vector field on M whose complete lift Xc to TM is constituted in a group of Crampin-Prince symmetries of the symplectic dynamics iζLwL=dEL canonically associated to L. Then

θL(Xc)isaconservationlaw{Xc(L)=0θL(ζL)isconstantonthe1-parameterorbitsofXc.

Proof. The assumption that the function θL(Xc) is a conservation law, means

ζL{θL(Xc)}=0,

In this way, by (3.1) we have

ζL{θL(Xc)}=LζL(iXcθL)=iXc(LζLθL).

Now, by the Cartan formula iζLd+diζL for the Lie derivative LζL along the field ζL, and the expression (2.3) above, we obtain

LζLθL=dL,
consequently
ζL{θL(Xc)}=iXc(dL)=XcL=0.

On the other hand, since by (3.3), the vector field Xc is the Hamiltonian vector corresponding to the function θL(Xc) the condition ζL {θL(Xc)} = 0 is equivalent to Xc(EL) = 0. In this way,

0=Xc(EL)=Xc(L-θL(ζL))=-XcθL(ζL).

Conversely, if Xc {θL(ζL)} = 0 and XcL = 0 then

Xc(L-θL(ζL))=0
which means ζL {θL(Xc)} = 0.

We shall finish this article with the following unexpected result.

Proposition 3.1.

With the previous notations, let us consider the Hamiltonian vector field Xc on TM,

iXcwL=d{θL(Xc)}.

Let Y be the unique vector field on TM such that iYwL = –θL Then

θL(Xc)=Y{θL(Xc)}.

Proof. We shall prove, in a more general manner, that for a Hamiltonian vector field Xf, that is iXfwL=df for some smooth function f on TM, we have

θL(Xf)=Yf.

Now

-θL(Xf)=iXf(iYwL)=-iY(iXfwL)=-iYdf=-Yf.

REFERENCES

[1]R Abraham and JE Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Company, 1978.
[7]MJ Gotay and JM Nester, Presymplectic Lagrangian systems. I. The constraint algorithm and the equivalence theorem, Annales de l’I.H.P. Physique théorique, Vol. 30, 1979, pp. 129-142.
[8]MJ Gotay and JM Nester, Presymplectic Lagrangian systems. II. The second order equation problem, Annales de l’I. H. P., section A, Vol. 32, 1980, pp. 1-13.
[10]J Kijowdki and WA Szczurba, Canonical structure for classical field theories, Comm. Math. Phys., Vol. 46, 1976, pp. 183-206.
[11]J Klein, Espaces variationels et mécanique, Ann. Inst. Fourier. (Grenoble), Vol. 12, 1962, pp. 1-124.
[12]J Klein, Opérateurs différentiels sur les variétés preque tangentes, C. R. Acad. Sci. (Paris), Vol. 257, 1963, pp. 2392-2394.
[13]O Krupková, The geometry of ordinary variational equations, Lecture Notes in Mathematics, 1678, Springer-Verlag, Berlin, 1997.
[15]G Prince, Toward a classification of dynamical symmetries in classical mechanics, Bull. Austral. Math. Soc., Vol. 27, 1983, pp. 53-71.
Journal
Journal of Nonlinear Mathematical Physics
Publication Date
2020/12
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.201203.001How to use a DOI?
Copyright
© 2020 The Author. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Javier Pérez Álvare
PY  - 2020
DA  - 2020/12
TI  - Orbits and Lagrangian Symmetries on the Phase Space
JO  - Journal of Nonlinear Mathematical Physics
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.201203.001
DO  - https://doi.org/10.2991/jnmp.k.201203.001
ID  - Álvare2020
ER  -