Variational Lie derivative and cohomology classes

TL;DR

This paper demonstrates that the variational Lie derivative trivializes certain cohomology classes, ensuring that conservation laws linked to symmetries are globally defined in the context of local variational problems.

Contribution

It establishes a connection between cohomology of local Lagrangians and the variational Lie derivative, showing the latter's cohomology class is zero, which has implications for conservation laws.

Findings

Cohomology class of the local variational Lie derivative is zero.

Conservation laws related to symmetries are globally defined.

Variational Lie derivative trivializes cohomology classes.

Abstract

We relate cohomology defined by a system of local Lagrangian with the cohomology class of the system of local variational Lie derivative, which is in turn a local variational problem; we show that the latter cohomology class is zero, since the variational Lie derivative `trivializes' cohomology classes defined by variational forms. As a consequence, conservation laws associated with symmetries ensuring the vanishing of the second variational derivative of a local variational problem are globally defined.