  The aim of this paper is to create a large geometrical background on the dual
1-jet space J^{1*}(T,M) for a multi-time Hamiltonian approach of the
electromagnetic and gravitational physical fields. Our geometric-physical
construction is achieved starting only from a given quadratic Hamiltonian
function of polymomenta H, which naturally produces a canonical nonlinear
connection N, a canonical Cartan N-linear connection C\Gamma(N) and their
corresponding local distinguished (d-) torsions and curvatures. In such a
context, we construct some geometrical electromagnetic-like and
gravitational-like field theories which are characterized by some natural
geometrical Maxwell-like and Einstein-like equations. Some abstract and
geometrical conservation laws for the multi-time Hamiltonian gravitational
physical field are also given.
