Variational symmetries and pluri-Lagrangian systems in classical mechanics

TL;DR

This paper explores the connection between pluri-Lagrangian systems and variational symmetries in classical mechanics, showing how multiple symmetries lead to multi-time Lagrangian structures with consistent equations.

Contribution

It establishes a method to construct pluri-Lagrangian 1-forms from systems with commuting variational symmetries and provides a Hamiltonian analogue for such constructions.

Findings

Constructed pluri-Lagrangian 1-forms from variational symmetries.

Multi-time Euler-Lagrange equations match original systems.

Extended the framework to Hamiltonian systems with commuting flows.

Abstract

We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with $m$ commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the $(m+1)$-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with $m$ commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpart of this construction, leading, for any system of commuting Hamiltonian flows, to a pluri-Lagrangian 1-form with coefficients depending on functions in the phase space.