Chaos in the Three Body Problem

Abstract

The purpose of this series of lecture notes is to give an outline of the basic tools required to show the occurrence of chaotic motions in the simplest non--integrable problems in Celestial Mechanics, such as the circular restricted planar 3--body problem. No formal proofs will be given here; they can be found in the references given for each section. Section 1 describes the linear and local theory of ordinary differential equations in the neighbourhood of a fixed point; the problems arising in the embedding of invariant stable and unstable manifolds are also discussed. Section 2 is about periodic orbits; the subjects discussed include variational equations, surfaces of section, the continuation of periodic orbits in the restricted 3--body problem, and bifurcation of hyperbolic periodic orbits from resonant periodic orbits. Section 3 covers fundamental models of resonance, the global behaviour of separatrices, and their intersections; all this allows to give at least an outline of the proof of the fundamental result --presented by Poincaré in his book Les méthodes nouvelles de la mécanique céleste-- by which homoclinic points must necessarily occur in the restricted problem. A short conclusion underscores the fact --shown in a rigorous vay much later-- that this in turn implies that chaos in the strongest possible sense occurs in the restricted problem, and is an essential feature of every non--integrable system, even very simple ones with only two degrees of freedom. I apologize for reporting here my lectures in a very short format, almost without comments in between the formulas and statements of the main results; my understanding of the purpose of this notes is that they should serve as a reminder of the existence of many subjects to be studied, rather than a complete presentation which could not be contained in this format.