From the restricted to the full three–body problem

Abstract

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three--body problerm. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first ap-proximation the product of the restricted three--body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem. For example, all the non--degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholom6 show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrailge point in the reduced three-body problem.