Chris Davison
Loughborough, Leicestershire, LE11 3TU, U. K.
Department of Mathematical Sciences
Loughborough University
Publications:
Davison C. M., Dullin H. R.
Geodesic Flow on ThreeDimensional Ellipsoids with Equal SemiAxes
2007, vol. 12, no. 2, pp. 172197
Abstract
Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semiaxes, here we study the remaining cases: Ellipsoids with two sets of equal semiaxes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semiaxes with $SO(2)$ symmetry, and ellipsoids with three semiaxes coinciding with $SO(3)$ symmetry. All of these cases are Liouvilleintegrable, and reduction of the symmetry leads to singular reduced systems on lowerdimensional ellipsoids. The critical values of the energymomentum maps and their singular fibers are completely classified. In the cases with $SO(2)$ symmetry there are corank 1 degenerate critical points; all other critical points are nondegenreate. We show that in the case with $SO(2) \times SO(2)$ symmetry three global action variables exist and the image of the energy surface under the energymomentum map is a convex polyhedron. The case with $SO(3)$ symmetry is noncommutatively integrable, and we show that the fibers over regular points of the energycasimir map are $T^2$ bundles over $S^2$.
