On the Carath\'eodory form in higher-order variational field theory

TL;DR

This paper generalizes the Carathéodory form from first-order to higher-order Lagrangians in field theory using geometric methods, providing a unified approach and discussing applications to general relativity.

Contribution

It introduces a geometric generalization of the Carathéodory form for second- and higher-order Lagrangians, extending previous results and unifying the theory.

Findings

Generalization of Carathéodory form to higher-order Lagrangians.

Coincidence with previous second-order results by Crampin and Saunders.

Application to the Hilbert Lagrangian in general relativity.

Abstract

The Carath\'eodory form of the calculus of variations belongs to the class of Lepage equivalents of first-order Lagrangians in field theory. Here, this equivalent is generalized for second- and higher-order Lagrangians by means of intrisic geometric operations applied to the well-known Poincar\'e--Cartan form and principal component of Lepage forms, respectively. For second-order theory, our definition coincides with the previous result obtained by Crampin and Saunders in a different way. The Carath\'eodory equivalent of the Hilbert Lagrangian in general relativity is discussed.