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Variational principles for locally variational forms. (English) Zbl 1110.58011

Summary: 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.

MSC:

58E30 Variational principles in infinite-dimensional spaces
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70S10 Symmetries and conservation laws in mechanics of particles and systems
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