Lepage Equivalents and the Variational Bicomplex

TL;DR

This paper develops a method to construct Lepage equivalents for Lagrangians of any order that satisfy a key closure property, using homotopy operators and vertical endomorphisms within the variational bicomplex framework.

Contribution

It introduces a new construction of Lepage equivalents satisfying the closure property using homotopy operators and explores the role of vertical endomorphisms in this process.

Findings

Constructed Lepage equivalents with the closure property for arbitrary order Lagrangians.

Used symmetric linear connections to extend local Lepage equivalents globally.

Investigated vertical endomorphisms for constructing homotopy operators and global tensors.

Abstract

We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper we investigate the r\^ole of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators.