Variational principles for locally variational forms

Abstract

We present the theory of higher order local variational principles in fibered manifolds, in which the fundamental global concept is a locally variational dynamical form. Any two Lepage forms, defining a local variational principle for this form, differ on intersection of their domains, by a variationally trivial form. In this sense, but in a different geometric setting, the local variational principles satisfy analogous properties as the variational functionals of the Chern–Simons type. The resulting theory of extremals and symmetries extends the first order theories of the Lagrange–Souriau form, presented by Grigore and Popp, and closed equivalents of the first order Euler–Lagrange forms of Haková and Krupková. Conceptually, our approach differs from Prieto, who uses the Poincaré–Cartan forms, which do not have higher order global analogues.