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Geodesic flow on three dimensional ellipsoids with equal semi-axes
preprint
posted on 2007-02-28, 12:48 authored by C.M. Davison, Holger R. DullinFollowing on from our previous study of the geodesic flow on three dimensional
ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellip-
soids with two sets of equal semi-axes with SO(2) × 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) sym-
metry 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 T2 bundles over S2.
History
School
- Science
Department
- Mathematical Sciences
Pages
436111 bytesPublication date
2007Notes
This is a pre-print.Language
- en