On the Structural Properties of Discrete-Time and Sampled-Data Hamiltonian Dynamics

While continuous-time Hamiltonian dynamics are naturally energy preserving with a symplectic flow, their discrete-time counterparts enhance either geometric or energy preservation properties, but rarely both within a unified framework. It is the object of this paper to more deeply investigate this question. In both linear and nonlinear settings, necessary and sufficient conditions characterizing discrete Hamiltonian dynamics that are conservative and symplectic are derived. The relationship with exact sampled models of continuous-time Hamiltonian dynamics are investigated, showing that such models, that preserve both energy and symplectic structures, do not generally fit into the proposed canonical form. Generalized Hamiltonian structures are, thus, introduced. On these bases, Hamiltonian integrators that preserve both the energy and the symplectic structure up to a prescribed order in the sampling period, are constructed. Some simulations on nonlinear test cases illustrate the theoretical findings.