Magnetic trajectories on 2-step nilmanifolds

TL;DR

This paper investigates magnetic trajectories on 2-step nilmanifolds, deriving solutions for invariant Lorentz forces, exploring examples in Heisenberg groups, and examining the existence of closed trajectories across energy levels.

Contribution

It provides explicit solutions for magnetic trajectories on 2-step nilpotent Lie groups and analyzes the existence of closed orbits, highlighting differences from exact form cases.

Findings

Solutions for magnetic trajectories on Heisenberg groups $H_3$ and $H_5$.

Presence of elliptic integral trajectories in $H_3$.

Conditions for closed magnetic trajectories across energy levels.

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.