Chaos, Regularity, and Noise in Self-Gravitating Systems

This paper summarises a number of new, potentially significant, results, obtained recently by the author and his collaborators, which impact on various issues related to the gravitational N-body problem, both Newtonianly and in the context of general relativity. Topics addressed include: (1) direct N-body simulations and their interpretation, with reference to the observed exponential instability towards small changes in initial conditions and the phenomenon of ``nonviolent relaxation;'' (2) the Hamiltonian structure of the collisionless Boltzmann equation of general relativity, i.e., the Vlasov-Einstein system; (3) ``transient ensemble dynamics,'' i.e.,,the short time statistical characterisation of collections of orbits in nonintegrable mean field potentials; and (4) the structural stability of the smooth potential approximation typically used in galactic dynamics.