The anisotropic Kepler problem in two dimensions

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 Poincare map for the trajectories is equivalent with a shift of th...