Finite dimensional Hamiltonian formalism for gauge and quantum field theories

We discuss in this article the canonical structure of classical field theory in finite dimensions within the pataplectic Hamiltonian formulation, where we put forward the role of Legendre correspondence. We define the generalized Poisson 𝔭-brackets which are the analogs of the Poisson bracket on forms. We formulate the equations of motion of forms in terms of 𝔭-brackets. As illustration of our formalism we present three examples: the interacting scalar fields, conformal string theory and the electromagnetic field.