Riemann Ellipsoids: Hamiltonian Formulation and Stability Analysis
The equilibria and stability of self-gravitating liquid masses has been studied and debated for more than a century by great physicists and mathematicians such as Newton, Maclaurin, Jacobi, Poincaré, Dirichlet, Riemann, and Chandrasekhar, and is still drawing interest from researchers today. Here I present an original approach to formulating the problem in the context of Hamiltonian theory, namely by applying moments of the position and velocity to the constrained Poisson bracket for a fluid. I then study the stability of a certain family of equilibrium ellipsoids with internal flow that depends linearly on the spatial coordinates (Riemann ellipsoids) using this constrained Hamiltonian formulation of the problem. This formulation allows us to use robust stability analysis methods, as well as study the dynamics in a straightforward way. The spectral stability results agree qualitatively with that of Chandrasekhar's, but the parameter value is slightly off, and the nonlinear stability analysis results do not give a definite answer due to the nature of the bifurcation (steady-state). It is still possible to use the Rayleigh-Ritz method to determine whether our system is nonlinearly unstable, but due to time constraints this was not done.