The gravitational N-body problem, a system of point masses moving under purely gravitational forces, poses a large number of challenges in mathematical physics, notwithstanding the simplicity with which the problem can be formulated. The absence of intrinsic length scales in Newtonian dynamics implies that singularities occur both at infinitely small distances between particles as well as at infinitely large separations. In high-energy physics, these singularities would be called ultraviolet and infrared divergencies, respectively. Here I list three areas of research in self-gravitating systems in which I have made some contributions.

### Topology

Since the two-body system can be solved analytically, the three-body problem is the simplest unsolved version of a self-gravitating system. Until electronic computers became available, work on the three-body problem was limited to the search for periodic orbits, and the derivation of perturbation expansions for applications to planetary orbits. When computers finally gave us a (virtual) laboratory, we could start to conduct gravitational scattering experiments. Apart from giving us quantitative information about the energy budgets of star clusters, these experiments also exhibited a variety of intriguing qualitative features, which I charted and analyzed in the paper

- The Topology of Three-Body Scattering, by Hut, P., 1983,
*Astron. J.***88**, 1549-1559.

A rather different example of the use of qualitative as well as quantitative techniques in the study of self-gravitating systems, in the limit of very large particle numbers, can be found in our paper

- Structure of Surfaces of Section, by Binney, J., Gerhard, O. & Hut, P., 1985,
*Mon. Not. R. astr. Soc.***215**, 59-65.

### Dynamical Instabilities

There are many examples known in stellar dynamics of equilibrium configurations, for which the density distributions do not change in time if the particles continue to follow their original orbits. Constructing such configurations is an interesting problem in itself, but determining their dynamical stability is more difficult. For the simplest case, of spherical density distributions, we combined numerical and analytic techniques in the paper:

- Dynamical Instabilities of Spherical Star Systems. I. Numerical Experiments for Generalized Polytropes, by Barnes, J., Goodman, J.J. & Hut, P., 1986,
*Astrophys. J.***300**, 112-131.

There we confirmed the existence of a radial instability found earlier, and we discovered two new nonradial instabilities.

### Exponential Instability: Lyaponov Coefficients

A completely different type of instability can be found on a `microscopic' level: if we perturb the orbit of even one particle in a self-gravitating system ever so slightly, the perturbation will affect the orbits of all other particles. These deviations will all grow exponentially fast. Even for the three-body problem, this road to chaos can be found in scattering experiments, as we described in our paper:

- Round-off Sensitivity in the
*N*-Body Problem, by Dejonghe, H. & Hut, P., 1986, in*The Use of Supercomputers in Stellar Dynamics*, eds. P. Hut and S. McMillan (Springer), pp. 212-218.

In the general gravitational N-body system, it turns out that the time scale for exponential growth is of order of a fraction of the crossing time, as we found in our paper:

- On the Exponential Instability of N-Body Systems, by Goodman, J., Heggie, D.C. & Hut, P., 1993,
*Astrophys. J.***415**, 715-733.

A connection with two-body relaxation was established in the paper

*Orbital Divergence and Relaxation in the Gravitational N-Body Problem*, by Hut, P. and Heggie, D.C., 2002, in the proceedings of the 84th Statistical Mechanics Conference (to celebrate the 65th birthdays of David Ruelle and Yasha Sinai),*J. Stat. Phys***xxx**, xxx-xxx (available in preprint form as astro-ph/0111015).

### Long-Term Evolution

Most $N$-body simulations do not extend far beyond the point of core-collapse. Following a system of self-gravitating point masses all the way down to its final evaporation is computationally very time-consuming. The presence of a tidal field will of course speed up the evaporation considerably, but it also introduces extra free parameters, and it complicates the theoretical analysis. The first detailed exploration of the long-term behavior of isoltad evaporating systems, without any tidal fields, was presented in

- Long-term evolution of isolated N-body sytems by Baumgardt, H., Hut, P. and Heggie, D.C., 2002,
*Mon. Not. R. astr. Soc.***336**, 1069-1081 (available in preprint form as astro-ph/0206258).