ON THE STABILITY AND NORMAL-MODES OF POLYTROPIC STELLAR-SYSTEMS USING THE SYMMETRIES OF LINEARIZED LIOUVILLES EQUATION

The stability and normal modes of oscillations of polytropic stellar systems are investigated using the symmetries of the linearized Liouville's equation. The O(3) symmetry of this linearized equation was utilized to separate the angle dependence of the eigenfunctions and hence to reduce the six dimensional phase-space problem to a two dimensional one in terms of magnitudes of position and momentum vectors. For the simplest mode of radial oscillations, the eigenvalue problem was solved numerically with a Rayleigh-Ritz variational scheme. Using 125 variational parameters, a high degree of convergence for the lowest eigenvalues was achieved. No negative eigenvalues were detected for any polytrope.