On the trajectories of O(1)-Kepler Problems

Abstract

The trajectories of the $\mathrm{O}(1)$-Kepler problem at level $n\ge 2$ are completely determined. It is found in particular 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. Moreover, it is shown that the group $\mathrm{GL}(n,\mathbb{R})/\mathrm{O}(1)$ acts transitively on both the set of oriented elliptic trajectories and the set of oriented parabolic trajectories. The method employed here is similar to the one used by Levi-Civita in the study of planar Kepler problem in 1920.

6 pages

References in corpus (3)

• Generalized Kepler Problems I: Without Magnetic Charges
• The Universal Kepler Problem
• Lorentz Group and Oriented MICZ-Kepler Orbits

Cited by in corpus (1)

• The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems