Two-dimensional harmonic oscillator and centered trochoïd curves: trajectories under constant magnetic field

Abstract

We studied the two-dimensional trajectories of a charged particle that was subjected to the simultaneous effect of a restoring force and a force perpendicular to its speed, which could thus be a magnetic force. Several results have been provided and compared with other modern works relative to this topic. In particular, using a classical method based on the introduction of complex variables, we showed that these trajectories are, when the force is attractive, a part of the centered trochoïd curves. When force was repulsive, we obtained a second family of trajectories, this time entirely original. In the two cases, several possible trajectories are detailed and limiting cases are studied. Possible applications are discussed, especially about the normal Zeeman effect and Paschen–Back effect.