The Hooke-Newton transmutation system of magnetic force

Abstract

Recently, it was reported that the Hooke-Newton transmutation of magnetic force can be generated by a conformal mapping in a uniform magnetic field. We perform the classical analysis of the transmutation system in this paper. First, the action variables are calculated and the energy is expressed in terms of the variables. The quantum spectrum of the system is manifested by the condition of the angular variables for the closed trajectories. Second, the trajectory equations are presented for the charge in the transmuted Coulomb field whose characteristics, attractive or repulsive, are determined by the signatures of both the angular momentum and the charge. It is shown that, given the released energy E and angular momentum pφ of the charged particle, the types of trajectories can be classified by the critical energy associated with the magnetic field strength . The feature makes the system only trap the particles with energy E < EC and the definite signatures of angular momentum and charge.