A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations: I. Covariant Hamiltonian Formalism

Several recent results on the Hamiltonian formalism in the calculus of variations are presented. In particular, I propose a new candidate for the covariant phase space and show that it carries a canonical multisymplectic structure. Corresponding covariant Legendre transformations are constructed; while not necessarily unique, the class of all such is completely characterized. A suitable notion of regularity is also defined. These results comprise the foundation of a truly Hamiltonian framework for the calculus of variations in general, and enable one to deal directly with higher order Lagrangians as well as multiple integrals in much the same way as one treats ordinary mechanics. The key ingredient in this work is a generalization of Kijowski and Szczyrba's notion of “multiphase space.”

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