Hamiltonian formalism for general variational problems

The present article introduces a generalization of the Hamiltonian field theory for a Lagrangian density, allowing the formulation of this kind of field theories for variational problem of more general nature than those associated to a classical variational problem. It is achieved by realizing that the usual construction of the Hamiltonian equations can be performed without the use of the so called Hamiltonian section, whose existence is problematic when general variational problems are considered. The developed formalism is applied to obtain an unified problem and a Hamiltonian field theory for the vakonomic Herglotz variational problem.