Geometry of variational partial differential equations and Hamiltonian systems
This is a survey of Hamiltonian field theory in jet bundles with a particular stress on geometric structures associated with Euler–Lagrange and Hamilton equations. Our approach is based on the concept of Lepage manifold, a fibred manifold endowed with a closed Lepage $(n+1)$-form where $n$ is the dimension of the base manifold, which serves as a background for formulation of a covariant Hamilton field theory related to an Euler–Lagrange form (representing variational equations), hence to the class of equivalent Lagrangians. Compared with conventional approaches, dependent upon choice of a particular Lagrangian, this is an important distinction which enables us to enlarge substantially the family of field Lagrangians which possess a canonical multisymplectic Hamiltonian formulation on the affine dual of the jet bundle, and can thus be treated without using the Dirac constraint formalism. Within the Hamiltonian theory on Lepage manifolds, the concepts of regularity and Legendre transformation are revisited and extended, and new formulas for the Hamiltonian and momenta are obtained. In this paper we focus on De Donder–Hamilton equations which arise from “short” (at most $2$-contact) Lepage $(n+1)$-forms. To illustrate the results we present regular Lepage manifolds (and the corresponding Hamiltonian formulation) for the Einstein and Maxwell equations.