I will discuss some recent results on the stability of spherically symmetric equilibria of several stellar models, including Euler-Poisson, Euler-Einstein and Vlasov-Einstein models. For Euler-Poisson and Euler-Einstein models, a turning point principle (TPP) for the sharp stability criterion is obtained. For both Vlasov-Einstein and Euler-Einstein models, the linear instability in the strong relativistic limit can be proved. To study the stability of these models, a combination of first order and 2nd order Hamiltonian formulations is used to derive the stability criterion and study the linearized equation for initial data in the energy space. This is joint work with Chongchun Zeng (on Euler-Poisson) and with Hadzic and Rein (on Euler-Einstein-Euler and Euler-Einstein).
