The fundamental Lepage form in two independent variables: a generalization using order-reducibility

TL;DR

This paper introduces a second-order generalization of the fundamental Lepage form for geometric calculus of variations on fibered manifolds with two variables, emphasizing order-reducibility and equivalence conditions.

Contribution

It develops a new second-order Lepage form that satisfies key equivalence and order-preservation properties, extending prior first-order results in field theory.

Findings

Provides a new second-order Lepage form satisfying key conditions

Completes previous attempts at second-order Lepage equivalents

Ensures the principal component extends the Poincaré-Cartan form

Abstract

A second-order generalization of the fundamental Lepage form of geometric calculus of variations over fibered manifolds with 2-dimensional base is described by means of insisting on (i) equivalence relation "Lepage differential 2-form is closed if and only if the associated Lagrangian is trivial" and, (ii) the principal component of Lepage form, extending the well-known Poincar\'{e}-Cartan form, preserves order prescribed by a given Lagrangian. This approach completes several attempts of finding a Lepage equivalent of a second-order Lagrangian possessing condition (i), which is well-known for first-order Lagrangians in field theory due to Krupka and Betounes.