On the trajectories of O(1)-Kepler Problems

TL;DR

This paper characterizes the trajectories of the O(1)-Kepler problem in higher dimensions, showing they are conic sections depending on energy, and explores symmetry group actions on these trajectories.

Contribution

It provides a complete description of trajectories for the O(1)-Kepler problem at level n≥2 and analyzes the symmetry group actions on elliptic and parabolic trajectories.

Findings

Non-colliding trajectories are ellipses, parabolas, or hyperbolas based on energy.

The symmetry group acts transitively on elliptic and parabolic trajectories.

Method parallels Levi-Civita's approach to planar Kepler problem.

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.