The three-body problem and the Lagrangian solutions

Abstract

The solution of the two-body problem is provided by Kepler’s laws, which state that for negative energies a point-mass moves on an ellipse whose focus coincides with the other point-mass. As shown by Poincaré [149], the dynamics becomes extremely complicated when you add the gravitational influence of a third body. In Section 4.1 we shall focus on a particular three-body problem, known as the restricted three-body problem, where it is assumed that the mass of one of the three bodies is so small that its influence on the others can be neglected (see, e.g., [21, 44, 94, 131, 163, 169]). As a consequence the primaries move on Keplerian ellipses around their common barycenter; a simplified model consists in assuming that the primaries move on circular orbits and that the motion takes place on the same plane. Action-angle Delaunay variables are introduced for the restricted three-body problem and the expansion of the perturbing function is provided.