The anisotropic Kepler problem in two dimensions

Abstract

The classical trajectories are investigated for a particle with an anisotropic mass tensor in an ordinary Coulomb potential. For negative energies (bound states) these trajectories are isomorphic with the geodesics on a Riemannian surface which can be immersed in a Euclidean space and which looks like a ``double snail.'' For vanishing energy (or near a collision) the equations of motion can be reduced to an autonomous system whose trajectories can be fully discussed. On the basis of extensive numerical computations, it has been possible to give a simple, yet complete description of all trajectories with negative energy. A binary sequence is associated with any trajectory where each term gives the sign of the position coordinate for the consecutive intersections with the ``heavy'' axis. If the binary sequence is represented by two real numbers, a one-to-one and continuous map from them to the initial conditions can be constructed. Thus, the Poincaré map for the trajectories is equivalent with a shift of the binary Bernoulli scheme (tossing a coin), and all the periodic orbits can be obtained systematically. A number of these are discussed to illustrate the consequences of the isomorphism with the binary sequences. Finally, the baker transformation and its use for finding the trajectories which connect any two given endpoints, is mentioned.