Geodesics on deformed spheres asymptotically described using the Funk transform

TL;DR

This paper studies geodesics on slightly deformed spheres, showing their asymptotic behavior can be described by a Hamiltonian system derived from the Funk transform, leading to an integrable phase space analysis.

Contribution

It introduces a novel approach using the Funk transform to describe geodesic dynamics on deformed spheres, revealing integrability and phase portrait topology.

Findings

Geodesic dynamics approximated by a Hamiltonian system

Hamiltonian derived via Funk transform of surface deviation

Trajectories characterized on the phase space sphere

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.