Dynamical stability of collisionless stellar systems and barotropic stars: the nonlinear Antonov first law

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.