Chapter 2 - The Kinematics of Rotations

Summary

Introduction

It is the purpose of this section to discuss the kinematics of rotations; by this term we mean the description of physical objects under rotation. Despite the apparent simplicity of the subject, the kinematics of rotations will prove to have some subtleties (see Sections 3 and 4).

The essential element in the description of physical systems lies in the association of a mathematical model to the underlying space (Mackey [1]). This association is necessarily postulated, and ultimately is an assumption as to the validity of a given system of relativity. Let us begin with the spatial concepts postulated in Newtonian relativity. The mathematical model to be associated with this physics is that physical (mass) points are to be identified with points belonging to a three-dimensional Euclidean space, E(3). “Three-dimensional”, means that a point is a triple of real numbers, point ↔ x ≡ (x₁, x₂, x₃), where, x_i ∈ ℝ “Euclidean space” means that under spatial symmetries belonging to the relativity group all distances between points are preserved (hence, all lengths and all angles are preserved).

The symmetries of Euclidean space can be composed from two special symmetries: (a) translations, which displace all points similarly: x → x' = x + a, a = (a₁, a₂, a₃); and (b) rotations and reversals, which leave one point, fixed – rotations preserve orientation, whereas reversals (Cartan's [2] term), reverse orientation.

Properties of Rotations

It is well known that a Euclidean symmetry leaving one point fixed leaves all points along some line through this point fixed.