Abstract

This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many presymplectic 2-forms used by Bridges [1997]. In this theory, solutions of a PDE are sections of a fiber bundle Y over a base manifold X of dimension n+1, typically taken to be spacetime. Given a connection on Y, a covariant Hamiltonian density H is then intrinsically defined on the primary constraint manifold PL, the image of the multisymplectic version of the Legendre transformation. One views PL as a subbundle of J 1 (Y) ⋆,theaffine dual of J 1 (Y), the first jet bundle of Y. A canonical multisymplectic (n+2)-form ΩH is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangian. Furthermore, we show that the n+1 2-forms ω (µ) defined by Bridges [1997] are a particular coordinate representation for a single multisymplectic (n+2)-form, and in the presence of symmetries, can be assembled into ΩH. A generalized Hamiltonian Noether theory is then constructed which

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