Magnetic equations on the Heisenberg group: symmetries, solutions and the inverse problem of the calculus of variations

TL;DR

This paper studies magnetic geodesics on the Heisenberg group, analyzing symmetries, explicit solutions, and their relation to variational problems, providing a comprehensive understanding of magnetic trajectories in this non-commutative setting.

Contribution

It offers explicit solutions for magnetic trajectories on the Heisenberg group and links them to variational principles, expanding understanding of magnetic equations in non-commutative geometry.

Findings

Explicit magnetic trajectories for invariant Lorentz forces

Symmetries enabling solution computation

Magnetic trajectories as solutions to variational problems

Abstract

The Heisenberg Lie group $H_3$ is modeled on the differentiable structure of $\mathbb{R}^3$ but equipped with another non-commutative product operation. By fixing the usual metric on the Heisenberg Lie group, this work provides a comprehensive overview of the behavior of magnetic geodesics for any invariant Lorentz force. After writing the magnetic equations, we found symmetries that enable the explicit computation of the magnetic trajectories for any homogeneous exact and non-exact magnetic form. Finally we show that these magnetic trajectories are solutions of a variational problem: we present explicit examples of Lagrangians.