Clebsch Canonization of Lie-Poisson Systems

TL;DR

This paper introduces the Clebsch canonization method to transform Lie-Poisson systems into canonical Hamiltonian systems, enabling structure-preserving numerical integration and providing new geometric insights.

Contribution

It presents a systematic coordinate and geometric procedure for Clebsch canonization, including the dual pair of momentum maps and their applications to Lie-Poisson systems.

Findings

Dual pair of momentum maps established for Lie-Poisson systems.

Canonical systems enable structure-preserving integrators like symplectic Runge-Kutta.

Examples include Kida vortex and heavy top systems.

Abstract

We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie-Poisson equation via a momentum map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. We also find another momentum map so that the pair of momentum maps constitute a dual pair under a certain condition. The dual pair gives a concrete realization of what is commonly referred to as collectivization of Lie-Poisson systems. It also implies that solving the canonized system by symplectic Runge-Kutta methods yields so-called collective Lie-Poisson integrators that preserve the coadjoint orbits and hence the Casimirs exactly. We give a couple of examples, including the Kida vortex and the heavy top on a movable base with controls, which are Lie-Poisson systems on $\mathfrak{so}(2,1)^{*}$ and $(\mathfrak{se}(3) \ltimes \mathbb{R}^{3})^{*}$, respectively.