On the trajectories of Sp(1)-Kepler problems

The classical Sp(1)-Kepler problems are formulated with the help of an idea from S. Sternberg. The trajectories of these models are determined via an idea originated from Levi-Civita. It is found that, for a non-colliding trajectory, its shadow-its projection to the external configuration space-is an ellipse, a parabola or a branch of hyperbola according as the total energy is negative, zero or positive. Moreover, it is shown that, for the Sp(1)-Kepler problems at level n≥2, the group SL(n,H)×R+ acts transitively on both the set of elliptic shadow trajectories and the set of parabolic shadow trajectories.