Theodore G. Northrop, Annals of Physics: 15: 79-101 (1961)

Title: The Guiding Center Approximation to Charged Particle Motion
DOI: https://doi.org/10.1016/0003-4916(61)90167-1

Abstract:
The equations governing the guiding center motion of a charged particle in an electromagnetic field are obtained simultaneously and deductively, without considering individually the special geometric situations in which one effect or another occurs alone. The general expression is derived for the guiding center velocity at right angles to the magnetic field B. This expression contains five terms arising in the presence of an electric field. They are in addition to the usual “E × B” drift. Because these terms are unfamiliar objects in the literature on plasmas, they are illustrated by simple examples. Three of the five drifts occur in rotating plasma machines of the Ixion type. One of these three is also shown to be responsible for the Helmholtz instability of a plasma. A fourth one gives the (low frequency) dielectric constant, while the fifth arises if the direction of B is time dependent. A detailed geometric picture of the fifth drift is given.

The equation governing the guiding center motion parallel to B is also derived for the general time-dependent field. The conditions are discussed under which it can be integrated into the form of an energy integral.

Finally the component of current density perpendicular to B in a collisionless plasma is shown to be the current due to the guiding center drift plus the perpendicular component of the curl of the magnetic moment per unit volume. Proofs of this have been given in the past for special cases, such as static fields, ∇ × B = 0, etc. This proof holds in general, provided conditions for adiabaticity are met. It is also true, but not proven in this paper, that the component of the current density parallel to B is the current due to the guiding center velocity parallel to B plus the parallel component of the curl of the magnetic moment per unit volume. A proper proof of the parallel component is quite lengthy.

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