Understanding rigid body motion in arbitrary dimensions

TL;DR

This paper presents a generalized, dimension-independent approach to rigid body motion using antisymmetric matrices and a novel tensor I, simplifying understanding and calculations in any number of dimensions.

Contribution

It introduces a dimension-agnostic framework for rigid body dynamics, replacing traditional three-dimensional constructs with antisymmetric matrices and a new tensor I.

Findings

Derived Euler equations in arbitrary dimensions.

Illustrated conserved quantities via Noether's theorem.

Simplified equations of motion for linked bodies.

Abstract

Why would anyone wish to generalize the already unappetizing subject of rigid body motion to an arbitrary number of dimensions? At first sight, the subject seems to be both repellent and superfluous. The author will try to argue that an approach involving no specifically three-dimensional constructs is actually easier to grasp than the traditional one and might thus be generally useful to understand rigid body motion both in three dimensions and in the general case. Specific differences between the viewpoint suggested here and the usual one include the following: here angular velocities are systematically treated as antisymmetric matrices, a symmetric tensor $I$ quite different from the moment of inertia tensor plays a central role, whereas the latter is shown to be a far more complex object, namely a tensor of rank four. A straightforward way to define it is given. The Euler equation is derived and the use of Noether's theorem to obtain conserved quantities is illustrated. Finally the equation of motion for two bodies linked by a spherical joint is derived to display the simplicity and the power of the method.