Symmetries of Helmholtz forms and globally variational dynamical forms

TL;DR

This paper explores the relationship between invariance properties of Helmholtz forms and their variational nature, demonstrating that local Euler--Lagrange forms are globally variationally equivalent.

Contribution

It establishes that the local system of Euler--Lagrange forms associated with invariant Helmholtz forms is globally variationally equivalent.

Findings

Invariance of Helmholtz forms implies local variationality.

Local Euler--Lagrange forms are globally equivalent.

Connects invariance properties with global variational structures.

Abstract

Invariance properties of classes in the variational sequence suggested to Krupka et al. the idea that there should exist a close correspondence between the notions of variationality of a differential form and invariance of its exterior derivative. It was shown by them that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form, so that the latter is locally the Euler--Lagrange form of a Lagrangian. We show that the corresponding local system of Euler--Lagrange forms is variationally equivalent to a global Euler--Lagrange form.