椭球上的测地线与单值测地线
Geodesics on the Ellipsoid and Monodromy
摘要：
椭球面上的测地线流方程是众所周知的，并且首先由雅可比在1838年通过分离哈密尔顿-雅可比方程的变量来求解。莫泽在1979年调查的情况下，一般椭球体与不同的半轴，并描述了一组积分这是不知道经典。在回顾了具有不同半轴的三维椭球上的测地线流的性质之后，我们研究了具有两个中间半轴相等的三维椭球，对应于一个旋转下的Hamilton不变量。该系统是刘维尔可积的，因此对应于能量动量映射的正则点的不变流形是3维环面。通过对能量-动量映射临界点的分析，给出了分岔图。我们找到了能量动量图临界值的纤维，并对作用变量进行了分析。我们证明了单值全局光滑作用变量存在的障碍是单值的。
英文摘要：
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.
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