Quaternions and Octonions in Mechanics

In fact, this group is Spin(3), the 2-fold cover of SO(3), the group of rotations of R. This has been known for quite some time and is perhaps the simplest realization of Hamilton’s expectations about the potential of quaternions for physics. One reason for the renewed interest is the fact that the resulting substitution of matrices by quaternions speeds up considerably the numerical calculation of the composition of rotations, their square roots, and other standard operations that must be performed when controlling anything from aircrafts to robots: four cartesian coordinates beat three Euler angles in such tasks. A more interesting application of the quaternionic formalism is to the motion of two spheres rolling on each other without slipping, i.e., with infinite friction, which we will discuss here. The possible trajectories describe a vector 2-distribution on the 5-fold S × S, which depends on the ratio of the radii and is completely nonintegrable unless this ratio is 1. As pointed out by R. Bryant, they are the same as those studied in Cartan’s famous 5-variables paper, and contain the following surprise: for all ratios different from 1:3 (and 1:1), the symmetry group is SO(4), of dimension 6; when the ratio is 1:3 however, the group is a 14-dimensional exceptional simple Lie group of type G2. The quaternions H and (split) octonions Os help to make this evident, through the inclusion S × S →֒ I(H)× H = I(Os). The distributions themselves can be described in terms of pairs of quaternions, a description that becomes “algebraic over Os” in the 1:3 case. As a consequence, Aut(Os), which is preciely that exceptional group, acts by symmetries of the system. This phenomenon has been variously described as “the 1:3 rolling mystery”, “a mere curiosity”, “uncanny” and “the first appearance of an exceptional group