Magnetic Trajectories on 2-Step Nilmanifolds

Abstract

The aim of this work is the study of magnetic trajectories on 2-step 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 explicit examples are computed in the Heisenberg Lie groups Hn for n= 3 , 5 , showing differences with the case of exact forms. Interesting magnetic trajectories related to elliptic integrals appear in H3. The question of existence of closed or periodic magnetic trajectories for every energy level on Lie groups or on compact quotients is treated. Conditions for the closeness of magnetic trajectories are proved.