Lyapunov's Stability Criteria for Plasmas

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Abstract
The orbit stability theory of Lyapunov has been adapted to the Vlasov‐Boltzmann equation governing plasmas. Both linear and nonlinear stability are considered. The theory is characterized by a search for Lyapunov functions, whose existence implies stability in analogy with particles trapped in a potential well, as in the energy principle. The most important result is an existence theorem for Lyapunov functions quadratic in perturbations in all linearly stable cases in which perturbations damp asymptotically (sufficiently fast). As a corollary, without damping, the existence of a quadratic Lyapunov function is necessary and sufficient to prevent exponential growth of perturbations. A prescription is given for finding Lyapunov functions which are constants of motion. An example is treated. The implication of nonlinear stability from linear stability with damping is discussed, and Dr. C. S. Gardner's direct proof of nonlinear stability of a Maxwellian plasma by Lyapunov's method is reported.

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