Discrete Hamiltonian Variational Integrators

TL;DR

This paper develops a theoretical framework for discrete Hamiltonian variational integrators, connecting continuous and discrete Hamiltonian mechanics, and introduces generalized integrators including symplectic methods, with invariance properties leading to a discrete Noether's theorem.

Contribution

It introduces a variational characterization of the exact discrete Hamiltonian and a class of generalized Galerkin Hamiltonian integrators, including symplectic Runge-Kutta methods.

Findings

Characterization of the exact discrete Hamiltonian.

Development of generalized Galerkin Hamiltonian integrators.

Establishment of a discrete Noether's theorem.

Abstract

We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators, which include the symplectic partitioned Runge-Kutta methods. We also characterize the group invariance properties of discrete Hamiltonians which lead to a discrete Noether's theorem.