Quaternions in Hamiltonian dynamics of a rigid body -- Part II. Relation of canonical Poisson and Lie-Poisson structures

TL;DR

This paper explores the relationship between Lie-Poisson and canonical Poisson structures in quaternion-based rigid body dynamics, revealing their equivalence and deepening understanding of the geometric structures involved.

Contribution

It demonstrates that the Lie-Poisson brackets for quaternion-based rigid body dynamics coincide with canonical brackets on the cotangent bundle, clarifying the geometric structure of the system.

Findings

Lie-Poisson brackets match canonical Poisson brackets

Symplectic structure of co-adjoint orbits relates to quaternion dynamics

Provides geometric insight into quaternion-based rigid body motion

Abstract

It was proposed the Lie group such that symplectic structure of orbits of co-adjoint representation of the group is revealed symplectic structure of a rigid body dynamics in quaternion variables. It is shown that Poisson brackets of corresponding Lie-Poisson structure coincide with canonical Poisson brackets on cotangent bundle of group unit quaternions.