On the closure property of Lepage equivalents of Lagrangians

TL;DR

This paper solves the open problem of constructing Lepage equivalents with the closure property for higher-order Lagrangians, providing a local yet often globally applicable solution relevant to physical theories.

Contribution

It introduces a general, local construction of Lepage equivalents with the closure property for all orders of Lagrangians, including a variant for reducible cases, often globally defined in physical applications.

Findings

Constructed Lepage equivalents are closed iff Euler-Lagrange expressions vanish.

Most of the constructed equivalents are globally defined in physically relevant cases.

For reducible second-order Lagrangians, the equivalents are of first order.

Abstract

Lepage equivalents of Lagrangians are a higher order, field-theoretical generalization of the notion of Poincare-Cartan form from mechanics and play a similar role: they give rise to a geometric formulation (and to a geometric understanding) of the calculus of variations. A long-standing open problem is the determination, for field-theoretical Lagrangians of order greater than one, of a Lepage equivalent with the so-called closure property: the Lepage equivalent is a closed differential form if and only if the Lagrangian has vanishing Euler-Lagrange expressions. The present paper proposes a solution to this problem, for general Lagrangians, of any order. The construction is a local one; yet, we show that in most of the cases of interest for physical applications, the obtained Lepage equivalent is actually globally defined. A variant of this construction, which is advantageous for reducible Lagrangians, is also introduced. In particular, for reducible Lagrangians of order two, the obtained Lepage equivalents are of order one.