Dynamical Instability and Statistical Behaviour of N-Body Systems

Abstract

We here describe the possibility of a synthetic description of the onset of Chaos in many degrees of freedom dynamical systems within the framework of the geometric description of dynamics. We show how this approach to instability helps to single out the transition between different regimes of stochasticity in typical as well peculiar N-body systems. It is shown how intermingled are the relationships between geometric properties of the manifold and stability properties of the dynamics, which do not follow any naive prescription. The implications concerning the possibility of a Statistical description are also addressed, and it is shown how the peculiarities of the interaction potential reflect themselves also on the geometric description. In particular, it is found that for the gravitational N-body system the dynamics possess features very similar to those of stable and tempered many body dynamical systems in the regime of strong stochasticity. All these issues are addressed from a semi-analytical approach, checked using well tailored numerical simulations, which help also to reinterpret some results recently appeared, and to correct to some extent previous claims presented in the literature.