Total Variation in Hamiltonian Formalism and Symplectic-Energy integrators

TL;DR

This paper develops a discrete total variation calculus within Hamiltonian formalism to derive high-order symplectic-energy integrators, connecting variational principles with numerical methods for Hamiltonian systems.

Contribution

It introduces a novel discrete variation calculus in Hamiltonian formalism and constructs symplectic-energy integrators of arbitrary order from a variational perspective.

Findings

Derived two-step symplectic-energy integrators of any finite order.

Established the relationship between direct symplectic integrators and variationally derived ones.

Provided a new variational framework for Hamiltonian numerical integration.

Abstract

We present a discrete total variation calculus in Hamiltonian formalism in this paper. Using this discrete variation calculus and generating functions for the flows of Hamiltonian systems, we derive two-step symplectic-energy integrators of any finite order for Hamiltonian systems from a variational perspective. The relationship between symplectic integrators derived directly from the Hamiltonian systems and the variationally derived symplectic-energy integrators is explored.