  We complete previous investigations on the dynamical stability of barotropic
stars and collisionless stellar systems. A barotropic star that minimizes the
energy functional at fixed mass is a nonlinearly dynamically stable stationary
solution of the Euler-Poisson system. Formally, this minimization problem is
similar to a condition of ``canonical stability'' in thermodynamics. A stellar
system that maximizes an H-function at fixed mass and energy is a nonlinearly
dynamically stable stationary solution of the Vlasov-Poisson system. Formally,
this maximization problem is similar to a condition of ``microcanonical
stability'' in thermodynamics. Using a thermodynamical analogy, we provide a
derivation and an interpretation of the nonlinear Antonov first law in terms of
``ensembles inequivalence'': a spherical stellar system with f=f(epsilon) and
f'(epsilon)<0 is nonlinearly dynamically stable with respect to the
Vlasov-Poisson system if the corresponding barotropic star with the same
equilibrium density distribution is nonlinearly dynamically stable with respect
to the Euler-Poisson system. This is similar to the fact that ``canonical
stability implies microcanonical stability'' in thermodynamics. The converse is
wrong in case of ``ensembles inequivalence'' which is generic for systems with
long-range interactions like gravity. We show that criteria of nonlinear
dynamical stability can be obtained very simply from purely graphical
constructions by using the method of series of equilibria and the turning point
argument of Poincare, as in thermodynamics.

Astrophysics