The Poincar\'e reduction problem for geodesics on deformed spheres

TL;DR

This paper investigates geodesics on hypersurfaces near spheres in n-dimensional space, using perturbation theory and integral geometry to reduce the problem to a Hamiltonian system on the Grassmann manifold, enabling topological classification.

Contribution

It introduces a Hamiltonian reduction of geodesic equations on deformed spheres using the X-ray transform, providing a framework for topological classification of geodesics.

Findings

Reduction to a Hamiltonian system on G(2,n)

Poisson brackets linked to Lie algebra of SO(n)

Topological classification achieved for 2D and 3D hypersurfaces

Abstract

We study geodesics on hypersurfaces close to the standard (n-1)-dimensional sphere in n-dimensional Euclidean space. Following Poincar\'e, we treat the problem within the framework of the analytical mechanics, and employ the perturbation theory with the view of obtaining a topological classification of the set of geodesics on a manifold. To that end we use the X-ray transform familiar in the integral geometry, and obtain the system of averaged equations of motion, which turns out to be a Hamiltonian one. The system serves an asymptotic reduction of the initial exact system of 2n-2 equations to that of 2n-4 equations on the Grassmann manifold G(2,n). The Poisson brackets of the reduction system are determined by the Lie algebra of the group SO(n). In the important cases of two-dimensional and a range of three-dimensional hypersurfaces it allows a topological classification of the set of geodesics.