The Cartan Form and Its Generalizations in the Calculus of Variations

Abstract

In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

Variational Sequences, Representation Sequences and Applications in Physics

“…Here we remind only some basic properties of Lepage n-forms, explored in the variational sequence theory. For more details and applications we refer the reader to the survey papers [54,59,60,67,76], and the book [55].…”

Frölicher-smooth geometries, quantum jet bundles and BRST symmetry

“…In classical Lagrangian field theory [34,29,32,33,38,39] one deals with arbitrary exterior forms on jet bundles of the configuration bundle. Furthermore one deals with the Lie bracket of arbitrary vector fields and the Frölicher-Nijenhuis bracket of vector-valued forms.…”

The fundamental Lepage form in variational theory for submanifolds

“…Replacing the initial Lagrangian by its Lepage equivalent, one obtains the same variational functional, but additionaly the geometric and variational properties of the functional are described by geometric operations acting on the corresponding Lepage equivalent. For a review of basic properties and results of the theory of Lepage forms in the calculus of variations, see Krupka, Krupková, and Saunders [17].…”