SYMMETRICAL PERIODIC-ORBITS OF THE MANY-BODY PROBLEM, RESONANCE AND PARADE OF PLANETS

The motion of a mechanical system consisting of n + 1 material points attracting one another according to Newton's law is investigated. A re versible system of differential equations is derived for the motion of n points relative to the ''main body''. A small parameter is introduc ed. When this parameter is equated to zero, each of the n points is at tracted by the ''main body'' only, and the generating system splits in to n two-body problems. Two types of generating periodic orbits, symme tric about the fixed set M of an automorphism, are considered: (1) wit h both eccentricities and inclinations equal to zero; (2) with inclina tions equal to zero. It is shown that such orbits can be continued to non-zero values of the small parameter, as a result of which the syste m has periodic solutions of the first and second kinds. Al these orbit s are resonant: the mean motions of the bodies relate to one another a s integers. In addition, at times that are multiples of the half-perio d the bodies are situated along a straight line, thus forming a ''para de of planets''. The results also apply to a ''Sun-planet-satellite'' type system. In the general theoretical part of the paper two methods are proposed for solving the problem of extending symmetric periodic m otions to non-zero parameter values, and an upper bound is estimated f or the domain of continuability.