  We analyse the problem of defining a Poisson bracket structure on the space
of solutions of the equations of motions of first order Hamiltonian field
theories. The cases of Hamiltonian mechanical point systems (as a (0 +
1)-dimensional field) and more general field theories without gauge symmetries
are addressed by showing the existence of a symplectic (and, thus, a Poisson)
structure on the space of solutions. Also the easiest case of gauge theory,
namely free electrodynamics, is considered: within this problem, a
pre-symplectic tensor on the space of solutions is introduced, and a Poisson
structure is induced in terms of a flat connection on a suitable bundle
associated to the theory.
