Lissajous-type orbits on a sphere and their non-Riemannian geometry

Abstract

A charged test particle is confined to the round sphere in a uniform ambient magnetic field. We reduce its motion to quadrature by elementary means and solve explicitly the family of trajectories which reaches the poles. On that family the azimuth advances at a constant rate and the latitude obeys a pendulum equation. The winding gained per oscillation is a complete elliptic integral of the first kind. That integral is a strictly increasing bijection onto the positive numbers. Every positive rational winding is therefore carried by exactly one value of the half-cyclotron frequency. Then we ask which geometry has these trajectories. No affine connection has them among its geodesics. A Finsler metric of Randers type has all of them. The threshold which separates the two dynamical regimes is precisely the condition for that metric to exist. The closed trajectories are its closed geodesics. The example places a non-Riemannian geometry within reach of an upper-level undergraduate or beginning graduate course in classical mechanics. We supply the derivations, the numerical recipes and the exercises.