  We consider stability of non-rotating gaseous stars modeled by the
Euler-Poisson system. Under general assumptions on the equation of states, we
proved a turning point principle (TPP) that the stability of the stars is
entirely determined by the mass-radius curve parameterized by the center
density. In particular, the stability can only change at extrema (i.e. local
maximum or minimum points) of the total mass. For very general equation of
states, TPP implies that for increasing center density the stars are stable up
to the first mass maximum and unstable beyond this point until next mass
extremum (a minimum). Moreover, we get a precise counting of unstable modes and
exponential trichotomy estimates for the linearized Euler-Poisson system. To
prove these results, we develop a general framework of separable Hamiltonian
PDEs. The general approach is flexible and can be used for many other problems
including stability of rotating and magnetic stars, relativistic stars and
galaxies.
