On the Trajectories of U(1)-Kepler Problems

Abstract

The classical $\mathrm{U}(1)$-Kepler problems at level $n\ge 2$ were formulated, and their trajectories are determined via an idea similar to the one used by Kustaanheimo and Stiefel in the study of Kepler problem. It is found that a non-colliding trajectory is an ellipse, a parabola or a branch of hyperbola according as the total energy is negative, zero or positive, and the complex orientation-preserving linear automorphism group of $\mathbb C^n$ acts transitively on both the set of elliptic trajectories and the set of parabolic trajectories.