On the Orbits of the Magnetized Kepler Problems in Dimension 2k+1

TL;DR

This paper classifies non-colliding orbits in magnetized Kepler problems in odd dimensions, showing they are conics with properties depending on energy, and describes the symmetry group actions on these orbits.

Contribution

It establishes that all non-colliding orbits are conics and characterizes their types based on energy, also identifying the symmetry group's transitive action on certain orbit sets.

Findings

All non-colliding orbits are conics.

Orbit type depends on total energy: ellipse, parabola, hyperbola.

The symmetry group acts transitively on elliptic and parabolic orbits.

Abstract

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the "external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.