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We study geodesics on hypersurfaces close to the standard (n-1)-dimensionalsphere in n-dimensional Euclidean space. Following Poincar\'e, we treat theproblem within the framework of the analytical mechanics, and employ theperturbation theory with the view of obtaining a topological classification ofthe set of geodesics on a manifold. To that end we use the X-ray transformfamiliar in the integral geometry, and obtain the system of averaged equationsof motion, which turns out to be a Hamiltonian one. The system serves anasymptotic reduction of the initial exact system of 2n-2 equations to that of2n-4 equations on the Grassmann manifold G(2,n). The Poisson brackets of thereduction system are determined by the Lie algebra of the group SO(n). In theimportant cases of two-dimensional and a range of three-dimensionalhypersurfaces it allows a topological classification of the set of geodesics.

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