Geodesics on deformed spheres asymptotically described using the Funk transform
Mathematical Physics
2010-03-30 v1 math.MP
Abstract
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.
Cite
@article{arxiv.1003.5449,
title = {Geodesics on deformed spheres asymptotically described using the Funk transform},
author = {D. O. Sinitsyn},
journal= {arXiv preprint arXiv:1003.5449},
year = {2010}
}