Geodetic Coils on Deformed Sphere

Abstract

We study geodetic lines on a surface generated by a small deformation of the standard 2D-sphere. We construct an auxiliary hamiltonian system with the view of describing geodetic coils and almost closed geodesics, by using the fact that loops of the coil can be well approximated by great circles of the sphere. The phase space of the auxiliary system is determined by the graph generated by separatrixes of its solutions, the vertices of the graph corresponding to almost closed geodesics and the edges to the geodetic coils joining them. Topological types of the graph depend on the parameters determining the deformation. Using the method of averaging in conjunction with the computer modelling of the auxiliary system, we obtain a fairly detailed visualization of geodesics on the deformed sphere.