A stability criterion for any collisionless stellar equilibrium and some concrete applications thereof

By viewing the Vlasov, or collisionless Boltzmann, equation of galactic dynamics as a Hamiltonian system with respect to an appropriate Lie bracket, an exact expression is derived for the energy associated with a phase-preserving, or symplectic, perturbation of an arbitrary equilibrium configuration, not assumed to possess any particular symmetries. Stability of the equilibrium hinges on the sign of the energy: positive energy for all perturbations implies linear and spectral stability. The existence of both negative and positive energy perturbations implies the existence of phase-preserving perturbations of zero linearized energy with arbitrarily large amplitudes. It is shown that a generic equilibrium depending on more than one constant of the motion will typically admit phase-preserving perturbations of negative energy that correspond to a local rearrangement of the velocity profile. It is also shown that, in analogy with spherical equilibria, a broad class of configurations with slab or cylindrical symmetries are stable with respect to centrally symmetric perturbations. 11 refs.