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.

Local variational problems and conservation laws

“…While conservation laws for equations without global Lagrangians have been studied by several authors, among them [6,8,7,17], a similar discussion, however, seems not to be in the literature and we hope it will clarify the highly involved situation somewhat.…”

On the Carathéodory Form in Higher-Order Variational Field Theory

“…In particular, the extremals are determined by means of the exterior derivative of the Lepage form. The meaning of Lepage forms for the calculus of variations and their basic properties, including examples of Lepage forms in geometric mechanics and field theory, have recently been summarized and further studied [13][14][15][16][17][18][19].…”

The Noether–Bessel-Hagen symmetry approach for dynamical systems

“…Our aim, in this paper, is to analyze Noether-type symmetries of dynamical systems with external forces, which leave invariant the corresponding equations of motion, known as the Noether-Bessel-Hagen symmetries [3]; for symmetries in local variational theories see monographs by Kossmann-Schwarzbach [4] and Krupka [5], and recent papers, e.g. [6][7][8][9][10][11][12]. As we will show, this is a straightforward generalization of the Noether Approach that can be extremely useful in several areas of physics like mechanics, field theory, cosmology and, in general, dynamical systems.…”