Constants of motion for the magnetic force: the angular momentum and the Laplace-Runge-Lenz vector

Abstract

It is well-known that an electric charge under a uniform magnetic field has a bidimensional motion if its initial position and velocity are perpendicular to this magnetic field. Although some constants of motion, as the energy and angular momentum, have been identified for this system, its features hide others. In this work, we build generalizations of the angular momentum and the Laplace-Runge-Lenz vector and show that these vectors are constants of motion. Moreover, from them, we find four dynamically independent conserved qualities.