Semi-classical quantization, adiabatic invariants and classical chaos

Einstein-Brillouin-Keller (E.B.K.) quantization of multidimensional non-separable classical hamiltonian dynamics is expeditiously carried out with classical adiabatic switching as hoped for in the early days of the old quantum theory by Ehrenfest, and reintroduced rather recently by Solov'ev. The method is briefly described along with a critique relating to the fact that the method seems to assume that invariant tori exist as a continuous one - parameter function of the coupling constant, although they do not. None the less, the method not only appears to work well for the types of systems envisaged by Ehrenfest, and Solov'ev, but continues to provide a useful asymptotic method following the onset of chaos. After distinguishing between adiabatic and geometric invariants, an analysis of adiabatic switching for the standard map is presented, with results given in the perturbative and strong coupling regimes. In the latter, a key phenomenological result is that principal non-adiabaticities can be understood in terms of the Farey organization of rationals spanning the relevant range of winding numbers. Finally the self-similar geometry of adiabatically generated `pseudoinvariant' curves is shown from the map of Siegal and Henon.