Phase plane analysis of the commensurable restricted three-body problem

All possible long period motions of the planar circular restricted three-body problem were studied by phase-plane analysis of the disturbing function when a commensurability relation exists. The short period argument was eliminated by an averaging process using an electronic computer. Initial conditions of periodic solutions are determined and the existence of asymmetric ones is proved for certain commensurabilities.The numerical computations performed cover the cases of large eccentricities and the curves of the mean value of the Hamiltonian were studied for the cases which correspond to collision periodic solutions.