Existence of oscillation modes in collisionless gravitating systems

We consider small perturbations on finite Antonov stable equilibrium configurations of the one-dimensional collisionless gravitating system, and radial perturbations on spherically symmetric three-dimensional systems. If we ignore the (time-dependent) gravitational field produced by the perturbation then 'differential motions' in phase space would lead to a phase-mixed equilibrium. We demonstrate that when we do take into account this self-consistent gravitational field, the dynamics change qualitatively - there can exist (real frequency) oscillation modes of the system which prevent the attainment of equilibrium by 'Landau damping'. (author)