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

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.