Stability of spherical stellar systems I : Analytical results

The so-called ``symplectic method'' is used for studying the linear stability of a self-gravitating collisionless stellar system, in which the particles are also submitted to an external potential. The system is steady and spherically symmetric, and its distribution function $f_0$ thus depends only on the energy $E$ and the squarred angular momentum $L^2$ of a particle. Assuming that $\partial f_0 / \partial E < 0$, it is first shown that stability holds with respect to all the spherical perturbations -- a statement which turns out to be also valid for a rotating spherical system. Thus it is proven that the energy of an arbitrary aspherical perturbation associated to a ``preserving generator" $\delta g_1$ [i.e., one satisfying $\partial f_0 / \partial L^2 \{ \delta g_1, L^2 \} = 0$] is always positive if $\partial f_0 / \partial L^2 \leq 0$ and the external mass density is a decreasing function of the distance $r$ to the center. This implies in particular (under the latter condition) the stability of an isotropic system with respect to all the perturbations. Some new remarks on the relation between the symmetry of the system and the form of $f_0$ are also reported. It is argued in particular that a system with a distribution function of the form $f_0 = f_0 (E,L^2)$ is necessarily spherically symmetric.