RESEARCH IN THE RESTRICTED PROBLEMS OF THREE AND FOUR BODIES.
Final technical rept.,
GENERAL PRECISION SYSTEMS INC LITTLE FALLS N J AEROSPACE RESEARCH CENTER
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The search for mathematical techniques sufficiently powerful to answer the difficult questions regarding stability of solutions of the nonlinear equations of motion in the restricted three-body problem has lead to a detailed investigation of the recent works of Kolmogorov, Arnold and Moser. These significant papers offer new constructive methods in the qualitative analysis of Hamiltonian systems, and relate problems of stability to the existence of periodic and almost periodic motions. For example, Arnolds work on the stability of motion in celestial mechanics resulted in his proof of the existence of almost periodic solutions in the n-body problem of planetary motion. Arnolds techniques enabled Leontovic to establish the stability of the triangular critical points of the planar restricted three-body problem. This theory can now be applied to the restricted problems of three and four bodies in the investigation of stability properties of known periodic motions. Two separate studies have been conducted in the restricted four-body problem. One dealt with the determination of particular solutions of the linearized equations of motion in the neighborhood of the L sub 1 libration point of the restricted three-body problem. The second part of the restricted four-body study has been concerned with the existence of periodic trajectories of the nonlinear equations of motion. The Poincare method of small parameters was successfully used to establish the existence of periodic trajectories in the neighborhoods of the libration points.