Geodesics on the Ellipsoid and Monodromy

TL;DR

This paper studies the integrable geodesic flow on a specific ellipsoid, analyzing the monodromy obstruction to globally defining smooth action variables, and extends classical results with new bifurcation and fiber analyses.

Contribution

It investigates the case of a three-dimensional ellipsoid with equal middle semi-axes, revealing monodromy as an obstruction to global action variables and analyzing bifurcation diagrams.

Findings

Liouville integrability of the system

Identification of monodromy as an obstruction

Detailed bifurcation diagram analysis

Abstract

The equations for 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 investigated the case of the general ellipsoid with distinct semi-axes and described a set of integrals which weren't know classically. After reviewing the properties of geodesic flow on the three dimensional ellipsoid with distinct semi-axes, we investigate the three dimensional ellipsoid with the two middle semi-axes being equal, corresponding to a Hamiltonian invariant under rotations. 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 maps gives the bifurcation diagram. We find the fibres of the critical values of the energy momentum map, and carry out an analysis of the action variables. We show that the obstruction to the existence of single valued globally smooth action variables is monodromy.