Historical Hamiltonian Dynamics: symplectic and covariant

Abstract: This paper presents a "historical" formalism for dynamical systems, in its Hamiltonian version (Lagrangian version was presented in a previous paper). It is universal, in the sense that it applies equally well to time dynamics and to field theories on space-time. It is based on the notion of (Hamiltonian) histories, which are sections of the (extended) phase space bundle. It is developed in the space of sections, in contradistinction with the usual formalism which works in the bundle manifold. In field theories, the formalism remains covariant and does not require a spitting of space-time. It considers space-time exactly in the same manner than time in usual dynamics, both being particular cases of the evolution domain. It applies without modification when the histories (the fields) are forms rather than scalar functions, like in electromagnetism or in tetrad general relativity. We develop a differential calculus in the infinite dimensional space of histories. It admits a (generalized) symplectic form which does not break the covariance. We develop a covariant symplectic formalism, with generalizations of usual notions like current conservation, Hamiltonian vector-fields, evolution vector-field, brackets, ... The usual multisymplectic approach derives form it, as well as the symplectic form introduced by Crnkovic and Witten in the space of solutions.