Collective Lie-Poisson integrators on $\mathbf{R}^{3}$

TL;DR

This paper introduces Lie-Poisson integrators for Hamiltonian systems on R^3 with the rigid body bracket, utilizing symplectic realization and Runge-Kutta schemes, also providing integrators for systems on the sphere S^2.

Contribution

It develops a novel class of Lie-Poisson integrators for R^3 systems using symplectic realization and Runge-Kutta methods, and derives integrators for spherical systems.

Findings

Effective Lie-Poisson integrators for R^3 demonstrated

Simple symplectic integrators for S^2 obtained

Method preserves geometric structure of Hamiltonian systems

Abstract

We develop Lie-Poisson integrators for general Hamiltonian systems on $\mathbf{R}^{3}$ equipped with the rigid body bracket. The method uses symplectic realisation of $\mathbf{R}^{3}$ on $T^{*}\mathbf{R}^{2}$ and application of symplectic Runge-Kutta schemes. As a side product, we obtain simple symplectic integrators for general Hamiltonian systems on the sphere $S^{2}$.