Classical Dynamics of Spinning Particles -- the Skew Product

Abstract

Since we are heading for questions originating in the field of quantum chaos – characterising properties of a quantum system by the degree of chaoticity or regularity, respectively, of the corresponding classical system -- two problems arise which we will duscuss in this chapter. Firstly, we have to deal with two dynamics rather than one, which, furthermore, are linked in one direction only, the translational dynamics in.uencing the spin dynamics but not the other way round. We would rather be dealing with only one classical counterpart instead, a dynamical system the chaotic properties of which in.uence the quantum mechanics. Secondly the spin dynamics is non-autonomous, but the properties we intend to talk about -- e.g. integrability or ergodicity -- are most conveniently dealt with in the language of flows. Therefore, we will now recombine spin and translational dynamics, which were separated in the semiclassical treatment, into one flow.