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

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.