On the relativistic motion of two charged particles in classical electrodynamics

Abstract

The relativistic, classical motion of a pair of charged particles interacting through their electromagnetic field is described by the equation of evolution for the density distribution function in the phase space of the particles. This equation is obtained (to second order in the charges) by the method of perturbation treatment of the Liouville equation for the system of particles plus field, previously used by Prigogine and Leaf to derive the Darwin Hamiltonian. Although this Hamiltonian is obtained to order ( v c ) 2, to higher orders there is no Hamiltonian for the pair of particles. The pair distribution function is, however, a constant of the motion and its equation of motion is a Liouville equation with a Hermitian Liouville operator. A one-particle Hamiltonian may be written for each particle when the motion of the other is prescribed. According to this description, each particle moves in an electromagnetic field whose potentials are the Coulomb gauge potentials produced by the other particle moving as a source with constant velocity.