# Orbits and Lagrangian Symmetries on the Phase Space

^{*}

^{*}Email: jperez@mat.uned.es

- 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
*w _{L}* 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 *w _{L}* 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

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*, *w _{L}*) is symplectic, the corresponding infinitesimal generator produce first integrals of the motion for the

*ζ*-dynamics whenever the

_{L}*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 *π _{M}*:

*TM*→

*M*the canonical projection. Given a coordinate neighborhood

*U*⊂

*M*with local coordinates (

*x*), let us denote with

_{i}*TU*the corresponding neighborhood in

*TM*coordinated by (

*x*,

_{i}*v*). Given

_{j}*x*∈

*M*, and

*y*∈

*T*, a vector

_{x}M*v*∈

*T*(

_{y}*TN*) such that

*X*on

*TN*is called vertical if

*X*is a vertical vector for each

_{y}*y*∈

*TN*. An element

*u*∈

*T*determines a vertical vector at any point

_{x}M*y*in the

*π*-fiber over

_{M}*x*, as the tangent vector at

*t*= 0 to the curve

*t*→

*y*+

*tu*. Naturally, we can define the vertical lift

*X*of a vector field

^{v}*X*on

*N*. Locally, if

*X*=

*X*, then

_{i}∂/∂x_{i}On the other hand, if *ϕ _{t}* 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

*Tϕ*of transformations of

_{t}*TM*whose generator is called the complete lift of

*X*to

*TN*and is denoted by

*X*. In local coordinates, if

^{c}*X*=

*X*then

_{i}∂/∂x_{i}Let us consider the canonical almost tangent structure *J* on *TM* locally defined by the (1, 1) tensor field

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

Let us consider a regular Lagrangian function
*x _{i}*,

*v*)

_{j}*TM*

*θ*=

_{L}*dL*◦

*J*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

*EL*=

*CL*−

*L*is the energy function corresponding to

*L*.

It is useful to consider the expression in local coordinates of the *Legendre* 1-form *θ _{L}*,

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

*a*are smooth functions (1 ≤

_{k}*k*≤

*n*). Consult Crampin [3] or Gotay and Nester [8]; also the monograph [6] goes through all the theory in its Chapter 7. This fact defines

*ζ*as a second-order differential equation field; the projections of its integral curves onto

_{L}*M*are the solutions of the system

*L*.

Finally, it easily follows from (2.2) the following expression for the *Legendre function* *θ _{L}*(

*ζ*) of the Lagrangian system

_{L}*L*,

## 3. *G*-ORBITS vs. *ζ*_{L}-ORBITS: CONSERVATIONS LAWS

_{L}

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

First of all we consider the Poisson structure associated to the symplectic phase space manifold (*TM*, *w _{L}*). For every

*f*∈

*C*

^{∞}(

*TM*), its Hamiltonian vector field is the unique vector field

*X*on

_{f}*TM*such that

*C*

^{∞}(

*TM*) is given by

It satisfies Leibniz’s rule

We say that *f* ∈ *C*^{∞}(*TM*) is a *conservation law* for the symplectic dynamics
*E _{L}* for the Poisson bracket. We denote by

*S*the set of all conservation laws

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
*dynamical symmetries* by Prince [15] and Crampin [4], are crystallized by vector fields
*ζ _{L}*,

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 *X ^{v}* and the complete lift

*X*of the vector field

^{c}*X*on

*M*, raise a theory of conserved quantities of a Lagrangian system, under the condition that

*df*on

*TM*and that

*X*= 0. Let us take as an example the conservation law

^{c}E_{L}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

In such a case, we say that
*TM* of Crampin-Prince symplectic symmetries.

As a direct consequence of this condition, we have

### Theorem 3.1.

*Let us consider a regular Lagrangian system
*

*Proof.* The assumption that the function *θ _{L}*(

*X*) is a conservation law, means

^{c}In this way, by (3.1) we have

Now, by the Cartan formula
*ζ _{L}*, and the expression (2.3) above, we obtain

On the other hand, since by (3.3), the vector field *X ^{c}* is the Hamiltonian vector corresponding to the function

*θ*(

_{L}*X*) the condition

^{c}*ζ*{

_{L}*θ*(

_{L}*X*)} = 0 is equivalent to

^{c}*X*(

^{c}*E*) = 0. In this way,

_{L}Conversely, if *X ^{c}* {

*θ*(

_{L}*ζ*)} = 0 and

_{L}*X*= 0 then

^{c}L*ζ*{

_{L}*θ*(

_{L}*X*)} = 0.

^{c}We shall finish this article with the following unexpected result.

### Proposition 3.1.

*With the previous notations, let us consider the Hamiltonian vector field X ^{c} on TM*,

Let *Y* be the unique vector field on *TM* such that *i _{Y}w_{L}* = –

*θ*Then

_{L}*Proof.* We shall prove, in a more general manner, that for a Hamiltonian vector field *X _{f}*, that is

*f*on

*TM*, we have

Now

## REFERENCES

### 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 -