Chapter 17: The Three-Dimensional Rotation Group

We study in this chapter some examples of classical mechanical realizations of the group R(3). We have already explained in Chapter 15 some of the important structural properties of this group and its Lie algebra. For applications to physics this is an important group, partly for its own sake and partly because it appears as a subgroup in the other space-time transformation groups For this reason, it is useful to include a discussion of the linear matrix representations of this group, although it would take us too far afield to prove all the interesting properties we mention. We also give a description of the connection between R(3) and the group SU(2) consisting of all complex unitary unimodular matrices in two dimensions. This is the group that describes the rotational behavior of “spinors” in contrast to the behaviors of vectors and tensors, which are adequately described by the group R(3). These matters, which are not exclusively related to classical mechanics, are taken up in the latter part of this chapter…