Introduction

One of the key features of the rigid body dynamics problems that we will shortly examine is the presence of a variable axis of rotation. This is one of the reasons for the richness of phenomena in rigid body dynamics. It is also a reason why this subject is intimidating. To quote the mechanician Louis Poinsot (1777–1859), from, “… if we have to consider the motion of a body of sensible shape, it must be allowed that the idea which we form of it is very obscure.” In this chapter, several representations of rotations are discussed that will enable us to establish both a clear picture of rigid body motions and straightforward proofs of several major results. To this end, many results on two key kinematical quantities for rigid bodies, rotation tensors and their associated angular velocity vectors, are discussed in considerable detail.

The subject of rotations in rigid body dynamics has a wonderful history, a wide range of interesting results, and an impressive list of contributors. Here, however, space limits the presentation of only the handful of results that are most relevant to our purposes. From a historical perspective, much of what is presented was established by Leonhard Euler (1707–1783) in his great works on rigid body dynamics that started to appear in the 1750s. The foundations Euler established were built upon by such notables as Cayley, Gauss, Hamilton, and Rodrigues in the early part of the 19th century.