Towards exact symplectic integrators from Liouvillian forms

TL;DR

This paper presents a new low order implicit symplectic integrator derived from Liouvillian forms, closely approximating Hamiltonian flow without specific Hamiltonian restrictions, based on a modified midpoint rule.

Contribution

It introduces a novel symplectic integrator using Liouvillian forms that improves flow approximation without special Hamiltonian assumptions.

Findings

The integrator is symmetric and closely follows Hamiltonian flow.

It is a modification of the midpoint rule based on an isotopy of Hamiltonian flow.

The method does not require particular hypotheses on the Hamiltonian.

Abstract

In this article we introduce a low order implicit symplectic integrator designed to follow the Hamiltonian flow as close as possible. This integrator is obtained by the method of Liouvillian forms and does not require particular hypotheses on the Hamiltonian. The numerical scheme introduced in this paper is a modification of the symplectic mid-point rule, it is symmetric and it is obtained by an isotopy of the deformation of the exact Hamiltonian flow to the straight line passing by two consecutive points of the discretized flow. This isotopy generates an alternative vector field on the flow lines transversal to the Hamiltonian vector field. We consider only the line arising from the mid-point to construct the symplectic integrator.