Quaternions in Hamiltonian dynamics of a rigid body -- Part I

TL;DR

This paper demonstrates how quaternion variables can be directly derived from canonical Poisson brackets in Hamiltonian mechanics of a rigid body, providing explicit quaternion-based motion equations.

Contribution

It introduces a method to obtain Poisson brackets in quaternion variables directly from canonical structures on cotangent bundles, linking quaternion representation to Hamiltonian dynamics.

Findings

Poisson brackets in quaternion variables derived from canonical brackets.

Quaternion parameters serve as dynamic variables in Hamiltonian mechanics.

Motion equations can be expressed through algebraic quaternion operations.

Abstract

This paper showed that Poisson brackets in quaternion variables can be obtained directly from canonical Poisson brackets on cotangent bundle of $SE(3)$ (or $SO(3)$) endowed by canonical symplectic geometry. Quaternion parameters in our case are just dynamic variables in canonical Hamiltonian mechanics of a rigid body on $T^*SE(3)$ The obtained results based on quaternions representation as explicit functions of rotation matrix elements of $SO(3)$ group. The relation of obtained Poisson structure to the canonical Poisson and symplectic structures on $T^*S^3$ were investigated. To derive the motion equations of Hamiltonian dynamics in quaternionic variables it is proposed to use the mixed frame of reference where translational degrees of freedom describes in the inertial frame of reference and degree of rotational freedom in the body frame. It turns out that motion equations for system with Hamiltonian of the rather general form can be written in algebraic operations on quaternions.