Bernoulli sequences and trajectories in the anisotropic Kepler problem

Abstract

The anisotropic Kepler problem is investigated in order to establish the one‐to‐one relation between its trajectories and the binary Bernoulli sequences. The Hamiltonian has a quadratic kinetic energy with an anisotropic mass tensor and a spherically symmetric Coulomb energy. Only trajectories in two dimensions with a negative energy (bound states) are discussed. The previous study of this system was based on extensive numerical computations, but the present work uses only analytical arguments. After a review of the earlier results, their relevance to the understanding of the relation between classical and quantum mechanics is emphasized. The main new result is to show the existence of at least one trajectory corresponding to each binary Bernoulli sequence. The proof employs a number of unusual mathematical tools, although they are all elementary. In particular, the virial as a function of the momenta (rather than the action as a function of the position coordinates) plays a crucial role. Also, different kinds of limiting trajectories, mostly involving a collision with the center of the Coulomb attraction are treated in some detail.