8 - RIGID BODIES

Summary

CHAPTER OVERVIEW

The motion of rigid bodies, also called rotational dynamics, is one of the oldest branches of classical mechanics. Interest in this field has grown recently, motivated largely by problems of stability and control of rigid body motions, for example in robotics (for manufacturing in particular) and in satellite physics. Our discussion of rigid-body dynamics will first be through Euler's equations of motion and then through the Lagrangian and Hamiltonian formalisms. The configuration manifold of rotational dynamics has properties that are different from those of the manifolds we have so far been discussing, so the analysis presents special problems, in particular in the Lagrangian and Hamiltonian formalisms.

INTRODUCTION

RIGIDITY AND KINEMATICS

Discussions of rigid bodies often rely on intuitive notions of rigidity. We want to define rigidity carefully, to show how the definition leads to the intuitive concept, and then to draw further inferences from it.

DEFINITION

A rigid body is an extended collection of point particles constrained so that the distance between any two of them remains constant.

To see how this leads to the intuitive idea of rigidity, consider any three points A, B, and C in the body. The definition implies that the lengths of the three lines connecting them remain constant, and then Euclidean geometry implies that so do the angles: triangle ABC moves rigidly. Since this is true for every set of three points, it holds also by triangulation for sets of more than three, so the entire set of points moves rigidly. In other words, fix the lengths and the angles will take care of themselves.