On reduction of the general three-body Newtonian problem and the curved geometry

Abstract

In the framework of an idea of separation of rotational and vibrational motions, we have examined the problem of reducing the general three-body problem. The class of differentiable functions allowing transformation of the 6D Euclidean space to the 6D conformal-Euclidean space is defined. Using this fact the general classical three-body problem is formulated as a problem of geodesic flows on the energy hypersurface of the bodies system. It is shown that when the total potential depends on relative distances between the bodies, three from six ordinary differential equations of second order describing the non-integrable hamiltonian system are integrated exactly, thus allowing reducing the initial system in the phase space to the autonomous system of the 6th order. In the result of reducing of the initial Newtonian problem the geometry of reduced problem becomes curved. The latter gives us new ideas related to the problem of geometrization of physics as well as new possibilities for study of different physical problems.