Abstract
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) × SO(2) symmetry, ellipsoids with equal larger or smaller semiaxes 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) × 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.
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Davison, C.M., Dullin, H.R. Geodesic flow on three-dimensional ellipsoids with equal semi-axes. Regul. Chaot. Dyn. 12, 172–197 (2007). https://doi.org/10.1134/S1560354707020050
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DOI: https://doi.org/10.1134/S1560354707020050