Stability Properties of the Riemann Ellipsoids

TL;DR

This paper investigates the stability of Riemann ellipsoids, demonstrating that most are stable over long timescales and expanding on previous findings with numerical evidence and Hamiltonian perturbation theory.

Contribution

It provides new numerical evidence that the stability regions of certain Riemann ellipsoids are larger than previously known and shows that almost all are Nekhoroshev-stable.

Findings

Regions of ellipticity are larger than previously identified.

Almost all Riemann ellipsoids are Nekhoroshev-stable.

Stability analysis uses Hamiltonian perturbation theory on a covering space.

Abstract

We study the ellipticity and the ``Nekhoroshev stability'' (stability properties for finite, but very long, time scales) of the Riemann ellipsoids. We provide numerical evidence that the regions of ellipticity of the ellipsoids of types II and III are larger than those found by Chandrasekhar in the 60's and that all Riemann ellipsoids, except a finite number of codimension one subfamilies, are Nekhoroshev--stable. We base our analysis on a Hamiltonian formulation of the problem on a covering space, using recent results from Hamiltonian perturbation theory.