Rotations in classical mechanics using geometric algebra

TL;DR

This paper presents a geometric algebra framework for describing rotations and rotational dynamics in classical mechanics, offering a unified and algebraic approach to analyze rigid body motion and gyroscope behavior.

Contribution

It introduces a novel formalism using rotors and bivectors in geometric algebra to analyze rotational dynamics, simplifying and unifying classical mechanics concepts.

Findings

Rotors effectively describe vector rotations in classical mechanics.

Bivectors provide a natural representation of angular velocity and momentum.

The formalism simplifies the derivation of Euler equations and energy expressions.

Abstract

In geometric algebra, the rotation of a vector is described using rotors. Rotors are phasors where the imaginary number has been replaced by a oriented plane element of unit area called a unit bivector. The algebra in three dimensional space relating vectors and bivectors is the Pauli algebra. Multivectors consisting of linear combinations of scalars and bivectors are isomorphic to quaternions. The rotational dynamics can be expressed entirely in the plane of rotation using bivectors. In particular, the Poisson formula providing the time derivative of the unit vectors of a moving frame are recast in terms of the angular velocity bivector and applied to cylindrical and spherical frames. The rotational dynamics of a point particle and a rigid body are fully determined by the time evolution of rotors. The mapping of the angular velocity bivector onto the angular momentum bivector is the inertia map. In the principal axis frame of the rigid body, the inertia map is characterised by symmetric coefficients representing the moments of inertia. The Huygens-Steiner theorem, the kinetic energy of a rigid body and the Euler equations are expressed in terms of bivector components. This formalism is applied to study the rotational dynamics of a gyroscope.