The Poincaré reduction problem for geodesics on deformed spheres

Details The Poincaré reduction problem for geodesics on deformed spheres The Poincaré reduction problem for geodesics on deformed spheres

2011-08-17

arxiv.org/abs/1108.3435v1

Physics

Mathematical Physics

In Russian, abstract in English, 14 pages, 4 figures

Scientific paper

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.

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