Variational derivatives in locally Lagrangian field theories and Noether--Bessel-Hagen currents

TL;DR

This paper develops a variational Cartan formula for the variational Lie derivative in the context of Krupka's variational sequence, and explores conditions under which Noether--Bessel-Hagen currents are equivalent to conserved Noether currents in invariant Lagrangian systems.

Contribution

It introduces a variational Cartan formula for the variational Lie derivative and characterizes when Noether--Bessel-Hagen currents correspond to conserved quantities in invariant theories.

Findings

Derived a variational Cartan formula applicable at any degree.

Established conditions for Noether--Bessel-Hagen currents to be variationally equivalent to Noether currents.

Showed that such currents, when they exist, are exact on-shell and generate conserved quantities.

Abstract

The variational Lie derivative of classes of forms in the Krupka's variational sequence is defined as a variational Cartan formula at any degree, in particular for degrees lesser than the dimension of the basis manifold. As an example of application we determine the condition for a Noether--Bessel-Hagen current, associated with a generalized symmetry, to be variationally equivalent to a Noether current for an invariant Lagrangian. We show that, if it exists, this Noether current is exact on-shell and generates a canonical conserved quantity.