Chris Davison

Loughborough, Leicestershire, LE11 3TU, U. K.
Department of Mathematical Sciences Loughborough University


Davison C. M., Dullin H. R.
Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semi-axes with $SO(2)$ symmetry, and ellipsoids with three semi-axes coinciding with $SO(3)$ symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum 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 non-degenreate. 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 energy-momentum map is a convex polyhedron. The case with $SO(3)$ symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are $T^2$ bundles over $S^2$.
Keywords: eodesic flow, integrable systems, symmetry, reduction, action variables
Citation: Davison C. M., Dullin H. R.,  Geodesic Flow on Three-Dimensional Ellipsoids with Equal Semi-Axes, Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 172-197

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