Variational Principle and Stability of Nonmonotonic Vlasov-Poisson Equilibria

The stability o f n onm onoton ic equilibria o f the V lasov-Poisson equation is assessed by using nonlinear constants o f m otion . The constants o f m otion m ake up the free energy o f the system , which upon variation y ields nonm onotonic equ ilibria . Such equilibria have not previously been obtainable from a variation principle, but here this is accom plished by the inclusion o f a passively advected tracer field. D efin iteness o f the second variation o f the free energy g ives a sufficient condition for stab ility in agreem ent w ith G ardner’s theorem [5], Previously, we have argued that indefin iteness im plies either spectral in stab ility or negative energy m odes, w hich are generically unstable w hen one adds d issipation or nonlinearity [6]. Such is the case for the non­ m onotonic equilibria considered.