Geodesic flow on three-dimensional ellipsoids with equal semi-axes |
| |
Authors: | C M Davison H R Dullin |
| |
Institution: | (1) Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK |
| |
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.
|
| |
Keywords: | geodesic flow integrable systems symmetry reduction action variables |
本文献已被 SpringerLink 等数据库收录! |
|