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 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 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, w_L) is symplectic, the corresponding infinitesimal generator produce first integrals of the motion for the ζ_L-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 π_M: TM → M the canonical projection. Given a coordinate neighborhood U ⊂ M with local coordinates (x_i), let us denote with TU the corresponding neighborhood in TM coordinated by (x_i, v_j). Given x ∈ M, and y ∈ T_xM, a vector v ∈ T_y(TN) such that πM*(v)=0 is called a vertical vector. A vector field X on TN is called vertical if X_y is a vertical vector for each y ∈ TN. An element u ∈ T_xM determines a vertical vector at any point y in the π_M -fiber over x, as the tangent vector at t = 0 to the curve t → y + tu. Naturally, we can define the vertical lift X^v of a vector field X on N. Locally, if X = X_i∂/∂x_i, then

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_t of transformations of TM whose generator is called the complete lift of X to TN and is denoted by X^c. In local coordinates, if X = X_i∂/∂x_i then

(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

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 L:TM→𝕉, that is, the Hessian matrix in every coordinate neighborhood (x_i, v_j)

is invertible. As a consequence, the 2-form on TM where 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

where 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

where a_k are smooth functions (1 ≤ 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_L as a second-order differential equation field; the projections of its integral curves onto M are the solutions of the system 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 θ_L(ζ_L) of the Lagrangian system L,

3. G-ORBITS vs. ζ_L-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, w_L). For every f ∈ C^∞(TM), its Hamiltonian vector field is the unique vector field X_f on TM such that iXfwL=df. In this way, the Poisson bracket on C^∞(TM) is given by

It satisfies Leibniz’s rule

and the Jacobi’s identity

We say that f ∈ C^∞(TM) is a conservation law for the symplectic dynamics iζLwL=dEL if it belongs to the centralizer of 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 L:TM→𝕉. These symmetries, denoted dynamical symmetries by Prince [15] and Crampin [4], are crystallized by vector fields X∈𝔛(M) that act as infinitesimal symmetries of the second order differential equation field 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^c 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 X^cE_L = 0. Let us take as an example the conservation law

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 X∈𝔛(M) conforms a vector field on TM of Crampin-Prince symplectic symmetries.

As a direct consequence of this condition, we have

We shall finish this article with the following unexpected result.