Geometric integration by parts and Lepage equivalents

TL;DR

This paper explores geometric integration by parts for contact forms, introduces a Residual operator, and generalizes the Krupka-Betounes equivalent to higher-order field theories.

Contribution

It compares contact form integration methods, defines a Residual operator for lower-degree forms, and extends the Krupka-Betounes equivalent to second order theories.

Findings

Defined a Residual operator for lower-degree contact forms.

Recovered the Krupka-Betounes equivalent for first order theories.

Discussed generalizations to second order field theories.

Abstract

We compare the integration by parts of contact forms - leading to the definition of the interior Euler operator - with the so-called canonical splittings of variational morphisms. In particular, we discuss the possibility of a generalization of the first method to contact forms of lower degree. We define a suitable Residual operator for this case and, working out an original conjecture by Olga Rossi, we recover the Krupka-Betounes equivalent for first order field theories. A generalization to the second order case is discussed.