CLASSICAL ORBITS IN POWER-LAW POTENTIALS

The motion of bodies in power-law potentials of the form V(r) = lambda r(alpha) has been of interest ever since the time of Newton and Hooke. Aspects of the relation between powers alpha and alphaBAR, where (alp ha + 2) (alphaBAR + 2) = 4, are derived for classical motion and the r elation to the quantum-mechanical problem is given. An improvement on a previous expression for the WKB quantization condition for nonzero o rbital angular momenta is obtained. Relations with previous treatments , such as those of Newton, Bertrand, Bohlin, Faure, and Arnold, are no ted, and a brief survey of the literature on the problem over more tha n three centuries is given.