The fundamental form of a homogeneous Lagrangian in two independent variables

TL;DR

This paper constructs a differential 2-form for homogeneous Lagrangians in two variables, which is closed if and only if the Lagrangian is null, extending properties known for first-order cases.

Contribution

It introduces a new differential 2-form for higher-order homogeneous Lagrangians in two variables, generalizing the fundamental Lepage equivalent concept.

Findings

The 2-form is closed exactly when the Lagrangian is null.

The construction applies to arbitrary order Lagrangians.

It parallels properties of first-order Lagrangians in jet bundle theory.

Abstract

We construct, for a homogeneous Lagrangian of arbitrary order in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the `fundamental Lepage equivalent' associated with first-order Lagrangians defined on jets of sections of a fibred manifold.