Periodic solutions of the N-body problem

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This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the ( N − 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N − 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N − 1 particles move approximately on a large circular orbit of the two body problems and the remaining N − 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.