High order symplectic partitioned Lie group methods

TL;DR

This paper presents a unified framework for constructing high-order symplectic integrators on cotangent bundles of Lie groups, extending classical Runge-Kutta methods to Lie group settings.

Contribution

It introduces a unified approach to derive high-order symplectic integrators on Lie groups from existing Lie group integrators, specifically using Runge--Kutta--Munthe-Kaas and Crouch--Grossman methods.

Findings

Able to obtain arbitrarily high order symplectic integrators

Unified approach applicable to different Lie group integrators

Extends classical methods to Lie group settings

Abstract

In this article, a unified approach to obtain symplectic integrators on T*G from Lie group integrators on a Lie group G is presented. The approach is worked out in detail for symplectic integrators based on Runge--Kutta--Munthe-Kaas methods and Crouch--Grossman methods. These methods can be interpreted as symplectic partitioned Runge--Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.