用Funk变换渐近描述的变形球面上的测地线
Geodesics on deformed spheres asymptotically described using the Funk transform
摘要：
我们考虑由标准二维球面的弱变形得到的曲面上的测地线。粒子在表面上的动力学可以用粒子角动量的平均演化来渐近描述。它示出，描述这种演变的系统具有一个哈密顿量，这是通过应用Funk变换的函数定义的表面从标准球的偏差。该系统的相空间为二维球面，因此它是可积的，其轨迹可以用球面上的相图进行拓扑描述。
英文摘要：
We consider geodesics on the surfaces obtained by weak deformations of the standard 2D-sphere. The dynamics of a particle on the surface can be asymptotically described by the averaged evolution of the particle's angular momentum. It is shown that the system describing this evolution has a Hamiltonian, which is obtained by applying the Funk transform to the function defining the deviation of the surface from the standard sphere. This system has the 2D-sphere as its phase space, so it is integrable and its trajectories admit of topological description in terms of its phase portrait on the sphere.
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