The charge distribution function in perturbation theory of classical electrodynamics

Abstract

In various problems of classical electrodynamics the solution is often found by solving the Boltzmann-Vlasov equation under proper constraints in six-dimensional phase space. A perturbation representation of the charge distribution function ψ is described in this paper. This representation is derived from the unperturbed distribution function ψ 0 by using a six-dimensional displacement vector ζ . It is shown that if ζ satisfies certain constraints, then ψ = [ exp(− ▽ · ζ)]ψ 0 satisfies the Boltzmann-Vlasov equation. The constraints which ζ must satisfy are dictated by none other than the Lorentz equation of motion of a charged particle. This constitutes one proof of the equivalence between the Boltzmann-Vlasov equation and the Lorentz equation. This proof may have advantageous aspects, especially in connection with perturbation calculations. As an example, the work of Lee, Mills and Morton on multipole oscillations of a throbbing beam is discussed in detail. A new and useful form for the equation of continuity is derived. The salient features of different kinds of phase spaces are compared by considering simple relativistic particle systems.