On the closure property of Lepage equivalents of Lagrangians

Abstract

Lepage equivalents of Lagrangians are a higher order, field-theoretical generalization of the notion of Poincaré-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 λ of order greater than one, of a Lepage equivalent Φλ with the so-called closure property:Φλ is a closed differential form if and only if λ has vanishing Euler-Lagrange expressions.The present paper proposes a solution to this problem, for general Lagrangians of order r≥1. The construction is a local one; yet, we show that in most of the cases of interest for physical applications, the obtained Lepage equivalent Φλ is actually globally defined.A variant of this construction, which is convenient in the cases when λ is a reducible Lagrangian, is also introduced. In particular, for reducible Lagrangians of order two, the obtained Lepage equivalents are of order one.