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 problem. 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 approximation 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–Bartholome 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 Lagrange point in the reduced three–body problem.