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We analyse the problem of defining a Poisson bracket structure on the spaceof solutions of the equations of motions of first order Hamiltonian fieldtheories. The cases of Hamiltonian mechanical point systems (as a (0 +1)-dimensional field) and more general field theories without gauge symmetriesare 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, apre-symplectic tensor on the space of solutions is introduced, and a Poissonstructure is induced in terms of a flat connection on a suitable bundleassociated to the theory.

The Geometry of the solution space of first order Hamiltonian field theories I: from particle dynamics to free Electrodynamics