## 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.

## Keywords

Periodic Orbit Invariant Manifold Unstable Manifold Stable Manifold Body Problem## Preview

Unable to display preview. Download preview PDF.

## References

- Abraham, R. and Marsden, J: 1967, ‘Foundations of Mechanics’, Benjamin, New YorkzbMATHGoogle Scholar
- Arnold, V.: 1976,
*Méthodes Mathématiques de la Mécanique Classique*, MIR, MoscouzbMATHGoogle Scholar - Froeschle, C.: 1984, ‘The Lyapounov Characteristic Exponents and Applications to the Dimension of the Invariant Manifolds of a Chaotic Attractor’, in
*Stability of the Solar System and Its Minor Natural and Artificial Bodies*, Szebehely, V. editor, Reidel, Dordrecht, 266–282Google Scholar - Hadjidemetriou, J.: 1982, ‘A qualitative study of stabilizing and destabilizing factors in planetary and asteroidal orbits’, in
*Applications of modern dynamics to celestial mechanics and astrodynamics*, Szebehely, V., editor, Reidel, Dordrecht, 25–44Google Scholar - Hartmann, P.: 1964,
*Ordinary differential equations*, J. Wiley and sonsGoogle Scholar - Henrard, J: 1990, ‘The adiabatic invsiriant in classical mechanics’.
*Dynamics Reported*, in pressGoogle Scholar - Hirsch, F. and Smale, S.: 1974,
*Differential equations, dynamical systems and linear algebra*. Academic presszbMATHGoogle Scholar - Lefschetz, S.; 1963
*,Differential equations: geometric theory*, Dover, New York (reprinted 1977)zbMATHGoogle Scholar - Message, P.J.: 1976, ‘Formal expressions for the motion of N planets in the plane, with the secular variations included, and an extension to Poisson’s theorem’, in
*Long-Time Predictions in Dynamics*, Szebehely and Tapley eds., Reidel Pu. Co., Dordrecht, Holland, 279–293Google Scholar - Milani, A.: 1988, ‘Secular perturbations of planetary orbits and their representation as series’, in
*Long Term Behaviour of Natural and Artificial N-Body Systems*, Roy, A.E., editor, Kluwer, Dordrecht, 73–108Google Scholar - Milani, A.: 1990, ‘Perturbation methods in Celestial Mechanics’, in Proceedings of the Goutelas Astronomy School, Froeschle, C., ed., in pressGoogle Scholar
- Newhouse, S.E.: 1980, ‘Lectures on Dynamical Systems’, in
*Dynamical Systems*, Marchioro, C., editor, liguori, Napoli, 209–311Google Scholar - Poincaré H.:
*Methodes Nouvelles de la Mechanique Celeste*, Vol. I, 1892; Vol. II, 1893; Vol. III, 1899; Gauthier-Villars, Paris (reprinted by Blanchard, Paris, 1987).Google Scholar