Geodesics on the ellipsoid and monodromy

After reviewing the properties of the geodesic flow on the three-dimensional ellipsoid, with distinct semi-axes, we investigate the three-dimensional ellipsoid with the two middle semi-axes equal, corresponding to a Hamiltonian invariant under rotations. The system is Liouville integrable, and symmetry reduction leads to a (singular) system on a two-dimensional ellipsoid with an additional potential and with a hard billiard wall inserted in the middle coordinate plane. We show that the regular part of the image of the energy-momentum map is not simply connected and there is an isolated critical value for zero angular momentum. The singular fibre of the isolated singular value is a doubly pinched torus multiplied by a circle. This circle is not a group orbit of the symmetry group, and thus analysis of this fibre is non-trivial. Finally we show that the system has a non-trivial monodromy, and consequently does not admit single-valued globally smooth action variables. (c) 2007 Elsevier B.V. All rights reserved.