Gauge theory of Riemann ellipsoids

Abstract

The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle . The structure group G = SO(3) is the vorticity group, and the bundle = GL+(3,) is the connected component of the general linear group. The base manifold is the space of positive-definite real 3×3 symmetric matrices, identified geometrically with the space of inertia ellipsoids. The angular momentum is not the only conserved quantity. The Kelvin circulation is also conserved as a consequence of gauge invariance. The bundle is a Riemannian manifold whose metric is determined by the kinetic energy. Nonholonomic constraints determine connexions on the bundle. In particular, the trivial connexion corresponds to rigid body motion, the natural Riemannian connexion to irrotational flow, and the invariant connexion to the falling cat.