Magnetic trajectories on 2-step nilmanifolds

Abstract

The aim of this work is the study of magnetic trajectories on nilmanifolds. The magnetic equation is written and the corresponding solutions are found for a family of invariant Lorentz forces on a 2-step nilpotent Lie group equipped with a left-invariant metric. Some examples are computed in the Heisenberg Lie groups $H_n$ for $n=3,5$, showing differences with the case of exact forms. Interesting magnetic trajectories related to elliptic integrals appear in $H_3$. The question of existence of closed or periodic magnetic trajectories for every energy level on Lie groups or on compact quotients is treated.