From quadratic Hamiltonians of polymomenta to abstract geometrical Maxwell-like and Einstein-like equations

Alexandru Oană, Mircea Neagu

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 the given quadratic Hamiltonian function $$ H=h_{ab}(t)g^{ij}(t;x)p^a_ip^b_j+U^{(i)}_{(a)}(t;x)p^a_i+\cal F(t;x) $$ which naturally produces a canonical nonlinear connection $N$, a canonical Cartan $N$-linear connection $C?(N)$ 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.