On the stability of motion of N-systems: a geometric approach
  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. 
Amr El-Zant