Variational principles and symmetries on fibered multisymplectic manifolds

TL;DR

This paper develops a comprehensive geometric framework using fibered multisymplectic manifolds to analyze variational principles, symmetries, and conservation laws in field theories and mechanics.

Contribution

It introduces a unified geometric approach to variational calculus, symmetries, and conservation laws applicable to a wide range of physical theories.

Findings

General geometric framework for variational calculus and symmetries

Inclusion of first and higher order field theories and mechanics

Extension of Noether's theorem in multisymplectic context

Abstract

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multisymplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.