Regular Reduction of Controlled Magnetic Hamiltonian System with Symmetry of the Heisenberg Group

TL;DR

This paper develops a regular point reduction method for controlled magnetic Hamiltonian systems with Heisenberg group symmetry, extending the theory and applying it to the motion of a Heisenberg particle in a magnetic field.

Contribution

It introduces a regular point reduction framework for CMH systems with Heisenberg group symmetry, including new equivalence concepts and detailed calculations on coadjoint orbits.

Findings

Established a regular point reduction theorem for CMH systems.

Derived reduced systems on generalized coadjoint orbits.

Applied the theory to analyze Heisenberg particle motion in magnetic fields.

Abstract

A controlled magnetic Hamiltonian (CMH) system is a regular controlled Hamiltonian (RCH) system with magnetic symplectic form, it is an important special case of RCH system. Note that there is a magnetic term on the cotangent bundle of the Heisenberg group, such that we can define a CMH system with symmetry of the Heisenberg group. Since the set of the CMH systems with symmetries is a subset of the RCH systems with symmetries, and it is not complete under the regular point reduction of RCH system, in this paper, then we give the regular point reduction of a CMH system with symmetry of the Heisenberg group, and discuss the M-CH-equivalence and MR-CH-equivalence, and prove the regular point reduction theorem for such CMH system. In particular, we deduce the regular point reduced CMH system on the generalization of coadjoint orbit of the Heisenberg group by calculation in detail. As an application, we consider the motion of the Heisenberg particle in a magnetic field.