High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition

TL;DR

This paper introduces high-order variational integrators on $SO(n)$ using polar decomposition, avoiding second derivatives of the exponential map, and demonstrates improved structure preservation and energy behavior in numerical simulations.

Contribution

It develops a novel high-order variational integrator on $SO(n)$ based on polar decomposition, enhancing structure preservation and computational efficiency.

Findings

Comparable efficiency to variational Runge--Kutta--Munthe-Kaas methods

Better preservation of Lie group structure

Improved near energy conservation

Abstract

A variational integrator of arbitrarily high-order on the special orthogonal group $SO(n)$ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second-order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie--Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on $SO(3)$ which demonstrate that they are comparable to variational Runge--Kutta--Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.