Selfgravitating gas spheres in a box and relativistic clusters: relation between dynamical and thermodynamical stability

We derive a variational principle for the dynamical stability of a cluster as a gas sphere in a box. Newtonian clusters are always dynamically stable, and for relativistic clusters, the relation between dynamical and thermodynamical instabilities is analyzed. The boundaries between dynamically and thermodynamically stable and unstable models are found numerically for relativistic stellar systems with different cutoff parameters. A criterion based on the binding energy curve is used for determination of the boundary of dynamical stability.