The guiding centre approximation in lowest order

Abstract

We present here a simple and direct derivation of the equations governing the motion of a charged particle in space- and time-dependent fields in the guiding centre approximation in lowest order. The method of attack is based initially on the expansion scheme of Bogolyubov & Zubarev (1955) combined with Northrop's averaging process over one gyration period. Thus, we first resolve the instantaneous velocity of the particle into three components, r = ê₁u + U_E + Tw, where u is the parallel component, U_E is the familiar E × B drift, and Tw is the instantaneous perpendicular component as measured in a frame moving with the U_E drift. Substituting this expression into the particle's (non-relativistic) vector equation of motion, resolving it into parallel and perpendicular components, and applying Northrop's averaging process, we obtain the averaged equations of motion, in lowest order, for the components u and w. From the latter equation we prove immediately that the magnetic moment of the particle is an adiabatic invariant to lowest order. Next, we resolve the instantaneous velocity of the guiding centre into two components, R = r — p, where p is a rapidly rotating vector directed from the guiding centre to the instantaneous position of the particle. Upon applying Northrop's averaging process we obtain at once the averaged velocity of the guiding centre including the complete set of first-order drifts. From this equation we readily deduce the energy gain equation to lowest order. It is then shown that the simple averaging prescription applied here is mathematically rigorous and yields correctly all lowest order results in an expansion in reciprocal powers of the cyclotron frequency.