Necessary and sufficient criteria for the stability of a hot particle ring-plasma system

The stability of a collisionless plasma with a population of energetic particles is analyzed. All species are assumed to obey the guiding center equations of motion, and thus, finite gyroradius and finite cyclotron frequency effects are not considered. The equilibrium magnetic field configuration is allowed to be arbitrary, and both equilibrium and perturbed electrostatic fields are included. A quadratic form is derived from the equations of motion and stability criteria are deduced from this form with the aid of Nyquist's technique. No apriori assumption concerning the constancy of the third adiabatic invariant is made. Nevertheless, in the case in which the drifts of the energetic particles are favorable (i.e. delta F/delta epsilon vertical bar/sub J/ < 0) marginal stability occurs at zero frequency and the quadratic form gives necessary and sufficient conditions for stability. The relationship of the instability thresholds predicted by the variational principle to those predicted by a modal analysis will be examined, and possible discrepancies between the two theories will be clarified