Nonlinear Stability of Stationary Plasmas - An Extension of the Energy-Casimir Method

We describe the time evolution of a nonrelativistic, collisionless plasma by the Vlasov–Poisson system. In [G. Rein, Math. Methods Appl. Sci., 17 (1994), pp. 1129–1140], the energy-Casimir method was used to prove nonlinear stability of steady states where the phase-space density of the particles is a decreasing function of the particle energy. In the present paper we extend this method to steady states with phase-space density depending on additional invariants of the particle motion. The existence of such steady states is established as well.