The problem of maximizing functionals in Newtonian stellar dynamics, and its relation to thermodynamic and dynamical stability

In an earlier paper, one of us studied the general problem of determining which Newtonian stellar systems maximize (or minimize) locally, at fixed energy and stellar masses, the functional given by the phase-space integral of a chosen function of the stellar distribution function. The criterion derived there for a maximum is incorrect. In this paper, a correct criterion is derived. Examples of the functionals associated with various types of equilibrium stellar systems are presented. In many cases, members of an appropriately constructed one-parameter sequence of spherical equilibria cease to maximize their associated functionals at the first maximum of the binding energy per unit mass along the sequence. Further, it is shown that an isotropic stellar system in collisionless equilibrium is dynamically stable to collisionless perturbations if it does maximize the unique (up to sign and constants) functional which it extremizes. For isothermal clusters, this implies a connection between dynamical stability and so-called thermodynamic stability: such a cluster is dynamically stable if it is thermodynamically stable in the sense that it locally maximizes the Boltzmann entropy. In many cases, it also implies a sufficient, bindingenergy criterion for dynamical stability: members of an appropriate spherical equilibrium sequence are dynamically stable to spherical collisionless perturbations at least up to the first maximum of the binding energy per unit mass along the sequence. Subject headings: instabilities - stars: stellar dynamics