Continuation of periodic solutions in three dimensions

Abstract

We use the methods of symplectic scaling and reduction to show that the reduced spatial three-body problem with one small mass is to the first approximation the product of the spatial restricted three-body problem and a harmonic oscillator. This allows us to prove that a nondegenerate periodic solution of the spatial restricted three-body problem can be continued into the reduced three-body problem with one small mass. The spatial three-body problem and the spatial restricted three-body problem admit two time-reversing symmetries. A solution which hits the fixed set of one of the symmetries at time 0 and the fixed set of the other at time T will be periodic of period 4 T and its orbit will be symmetric with respect to both symmetries. Such solutions are called doubly symmetric. We prove that a nondegenerate doubly symmetric periodic solution of the spatial restricted three-body problem can be continued into the reduced three-body problem with one small mass.