  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.
