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

Gabriela Ovando, Mauro Subils

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.

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

Abstract

The Heisenberg Lie group is modeled on the differentiable structure of 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.
Paper Structure (5 sections, 10 theorems, 89 equations, 1 table)

This paper contains 5 sections, 10 theorems, 89 equations, 1 table.

Table of Contents

  1. Introduction
  2. Symmetries for the magnetic equation
  3. The magnetic equation on the Heisenberg Lie group
  4. Magnetic trajectories that are geodesics
  5. The inverse problem of the calculus of variations

Key Result

Lemma 2.1

Let $(M,g)$ be a Riemannian manifold, and let $F$ denote a Lorentz force on $M$. Consider the group $H=\operatorname{Iso}(M)\times\mathbb{R}^{*}$.

Theorems & Definitions (29)

  • Lemma 2.1
  • Example 2.2
  • Corollary 2.3
  • Example 2.4
  • Proposition 2.5
  • Remark 2.6
  • Lemma 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • ...and 19 more