Stability properties of the Riemann ellipsoids

Abstract: The Riemann ellipsoids are steady motions of an ideal, incompressible, self-gravitating fluid that retain an ellipsoidal shape. The existence, properties, and stability of these steady motions have been investigated since Newton's time. Most of the stability results are due to Riemann, who studied Lyapunov stability, and to Chandrasekhar, who studied (primarily numerically) spectral stability, thus obtaining Lyapunov instability results. This article addresses the "Nekhoroshev stability" (stability for finite, but very long time scales) of those Riemann ellipsoids that are spectrally stable but of unknown Lyapunov stability. We base our analysis on a Hamiltonian formulation of the problem derived from Riemann's original formulation (which we interpret here as a formulation on a covering space) using recent results from Hamiltonian perturbation theory. Given the complexity of the system, we resort to numerical calculations at certain steps of the stability analysis. As a prerequisite to our analysis, we repeat the Lyapunov and spectral stability analyses, finding important discrepancies with Chandrasekhar's findings. We provide numerical evidence that (i) There are spectrally stable ellipsoids of type II and the region of spectral stability of the ellipsoids of type III is significantly larger than that found by Chandrasekhar. The regions of spectral stability of the ellipsoids of types I, II and III have a finer and subtler structure than was previously believed. (ii) All Riemann ellipsoids, except a finite number of codimension-one resonant subfamilies, are Nekhoroshev-stable.