Rigid Body Dynamics in a Constant Curvature Space and the ‘1D-up’ Approach to Conformal Geometric Algebra

Abstract

We discuss a ‘1D up’ approach to Conformal Geometric Algebra, which treats the dynamics of rigid bodies in 3D spaces with constant curvature via a 4D conformal representation. All equations are derived covariantly from a 4D Lagrangian, and definitions of energy and momentum in the curved space are given. Some novel features of the dynamics of rigid bodies in these spaces are pointed out, including a simple non-relativistic version of the Papapetrou force in General Relativity. The final view of ordinary translational motion that emerges is perhaps surprising, in that it is shown to correspond to precession in the 1D up conformal space. We discuss the alternative approaches to Euclidean motions and rigid body dynamics outlined by Gunn in Chap. 15 and Mullineux and Simpson in Chap. 17 of this volume, which also use only one extra dimension, and compare these with the Euclidean space limit of the current approach.