1999], Symmetries of Riemann ellipsoids

Abstract: The results of Dirichlet, Dedekind and Rie-mann on 'ellipsoidal figures of equilibrium ' of rotating self-gravitating fluids are reviewed in the context of the geomet-ric theory of Hamiltonian systems with symmetry. In par-ticular Riemann's classification is derived using only the ex-istence of physically natural rotational symmetries, and so is shown to be applicable to models of liquid drops, atomic nuclei and elastic bodies as well as self-gravitating fluids. Similarly Dedekind's transposition symmetry is obtained as a simple consequence of the rotational symmetries. A detailed descrip-tion is given of a generalization of 'self-adjoint ' ellipsoids. The symmetry groups of the different types of ellipsoidal figures of equilibrium are also computed. Key words: Riemann ellipsoids, conserved quantities, momentum maps, affine rigid body, relative equilibrium