DIVERGENCE OF NEARBY TRAJECTORIES FOR THE GRAVITATIONAL N-BODY PROBLEM

This paper considers the Hamiltonian flow of a collection of N self-gravitating Newtonian point masses, viewed as a geodesic flow on an appropriate curved but conformally flat 3N-dimensional manifold. It is proved that, with respect to the natural Euclidean measure, the probability that a random perturbation δqa of a random geodesic ua with comparable kinetic and potential energies will feel a positive curvature K(u, δq) decreases exponentially to zero as N → ∞. This suggests, but unfortunately does not prove, that at least for short times, large self-gravitating systems should exhibit a "mixing-type" or "chaotic" behavior.