posted on 2018-10-26, 09:13authored byChristopher M. Davison
The equations for the geodesic flow on the ellipsoid are well known, and were first
solved by Jacobi in 1838 by separating the variables of the Hamilton–Jacobi equation. In
1979 Moser showed that the equations for the geodesic flow on the general ellipsoid with
distinct semi-axes are Liouville-integrable, and described a set of integrals which weren't
known classically. These integrals break down in the case of coinciding semi-axes.
After reviewing the properties of the geodesic flow on the three-dimensional ellipsoid
with distinct semi-axes, the three-dimensional ellipsoid with the two middle semi-axes being
equal, corresponding to a Hamiltonian invariant under rotations, is investigated, using
the tools of singular reduction and invariant theory. The system is Liouville-integrable
and thus the invariant manifolds corresponding to regular points of the energy momentum
map are 3-dimensional tori. An analysis of the critical points of the energy momentum
map gives the bifurcation diagram. The fibres of the critical values of the energy momentum
map are found, and an analysis is carried out of the action variables. The obstruction
to the existence of single valued globally smooth action variables is monodromy. [Continues.]
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Publication date
2006
Notes
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy at Loughborough University.