A distinguished Riemannian geometrization for quadratic Hamiltonians of polymomenta

TL;DR

This paper develops a specialized Riemannian geometric framework on the dual 1-jet space tailored for multi-time quadratic Hamiltonian functions, integrating connections and curvature structures derived from the Hamiltonian.

Contribution

It introduces a novel Riemannian geometrization on the dual jet space specifically for multi-time quadratic Hamiltonians, including a nonlinear connection and a generalized Cartan connection.

Findings

Constructed a distinguished Riemannian geometrization for quadratic Hamiltonians.

Defined a nonlinear connection and a generalized Cartan N-linear connection.

Derived local d-torsions and d-curvatures from the Hamiltonian function.

Abstract

In this paper we construct a distinguished Riemannian geometrization on the dual 1-jet space J^{1*}(T,M) for the multi-time quadratic Hamiltonian functions. Our geometrization includes a nonlinear connection N, a generalized Cartan canonical N-linear connection (together with its local d-torsions and d-curvatures), naturally provided by a given quadratic Hamiltonian function depending on polymomenta.