  Much of standard galaxy dynamics rests on the implicit assumption that the
corresponding N-body problem is (near) integrable. This notion although leading
to great simplification is by no means a fact.
  It is therefore important to develop methods characterizing the stability of
motion of N-body systems. A method based on the geometry of a system's phase
space is examined and compared with more standard techniques.
  We also present the results of high precision N-body simulations of the
dynamics of systems of 231 point particles over a few dynamical times. The
scalar and Ricci (or mean) curvatures are calculated along the trajectories and
the relative instabilities of different systems are thus derived along with the
corresponding ``mixing'' timescales. The effect of softening is also examined.
We compare the predictions obtained from the calculations above with the
spatial evolution of the different systems and deduce that this is well
described.
