Symmetries, Newtonoids vector fields and conservation laws in the Lagrangian $k$-symplectic formalism

TL;DR

This paper extends the classical Noether's Theorem to the $k$-symplectic formalism, establishing links between symmetries, conservation laws, and vector fields for systems with multiple variables, and provides new proofs and examples.

Contribution

It introduces a new proof of Noether's Theorem for $k$-symplectic systems and explores the conditions under which the converse holds, expanding the theoretical framework.

Findings

Cartan symmetries induce conservation laws in $k$-symplectic formalism

The converse of Noether's Theorem holds under certain assumptions

Examples illustrate when symmetries and conservation laws are related or not

Abstract

In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether's Theorem and its converse, in the framework of the $k$-symplectic formalism, using the Fr\"olicher-Nijenhuis formalism on the space of $k^1$-velocities of the configuration manifold. For the case $k=1$, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's Theorem and its converse. For the case $k>1$, we provide a new proof for Noether's Theorem, which shows that, in the $k$-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's Theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.