Variationally equivalent problems and variations of Noether currents

TL;DR

This paper explores the relationship between variational problems, symmetries, and conserved currents, showing how certain currents are variationally equivalent and conditions for their global conservation.

Contribution

It establishes a link between generalized symmetries and Noether currents, providing conditions for their global conservation in local variational problems.

Findings

Conserved current associated with a generalized symmetry is variationally equivalent to the variation of the strong Noether current.

A sufficient condition for the current to be global is identified.

The work applies to systems with non-vanishing cohomology classes.

Abstract

We consider systems of local variational problems defining non vanishing cohomolgy classes. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order such a current be global.