Periodic orbits in gravitational systems

Abstract

The periodic orbits play an important role in the study of the stabil- ity of a dynamical system. The methods of study of the stability of a periodic orbit are presented both in the general case and for Hamil- tonian systems. The Poincare map on a surface of section is presented as a powerful tool in the study of a dynamical system, especially for two or three degrees of freedom. Special attention is given to nearly integrable dynamical systems, because our solar system and the extra solar planetary systems are considered as perturbed Keplerian systems. The continuation of the families of periodic orbits from the unper- turbed, integrable, system to the perturbed, nearly integrable system, is studied. 1. The gravitational N -Body problem The Newtonian gravitational force is the dominant force in the N - Body systems in the universe, as for example in a planetary system, a planet with its satellites, or a multiple stellar system. The long term evolution of the system depends on the topology of its phase space and on the existence of ordered or chaotic regions. The topology of the phase space is determined by the position and the stability character of the periodic orbits of the system (the fixed points of the Poincare map on a surface of section). Islands of stable motion exist around the stable periodic orbits. chaotic motion appears at unstable periodic orbits. This makes clear the importance of the periodic orbits in the study of the dynamics of such systems.