On the orbits of magnetized Kepler problems in dimension 2k+1

Abstract

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" R 2k+1 \ {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 SO + (1, 2k + 1) × R + acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.

Particle Motion in Monopoles and Geodesics on Cones

“…Taking the norm, we see that L = 0 if and only if λ = 0 and 9 r is everywhere parallel to r. This corresponds to motion at constant speed on a straight line that passes through the origin. Since those curves will come out often here and in subsequent sections, we give them the following name (a term borrowed from [1]). R n such that 9 r is everywhere parallel to r.…”

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“…So it suffices to understand the projection of the orbit onto R 2k+1 * . This projection curve was found [1] to be either a part of straight line (colliding orbit) or a conic (non-colliding orbit).…”

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