On the lowest energy state of a collisionless self-gravitating system under phase space volume constraints

The problem of minimizing the energy of a collisionless self-gravitating system under the only constraint of phase space volume conservation is reconsidered. With respect to a recent paper on the same subject, we bring about essentially three new results: (i) we establish a lower bound on the energy by using arguments which are both simpler and more general than those of these authors; (ii) we prove that the lowest energy state is necessarily spherically symmetric (in the physical space), and (iii) we show that, if we fix also the total angular momentum J of the system, then there is no lowest energy state satisfying this further constraint if J not= 0. (author)