Gregory Buck, Celestial Mechanics

Celestial Mechanics
Most smooth closed space curves contain approximate solutions of the n-body problem Nature, 395, 3 Sept 1998, 51-53
In this paper a large class of approximate solutions to the n-body problem is described: the masses are equidistributed along closed space curves and move with uniform speed along the curves. Every generic smooth closed space curve contains a family of these approximate solutions. The theory also holds for sets of closed curves, therefore there is an infinite set of approximate solutions for every knot and link type. The approach is closely related to constructions in vortex dynamics and physical knot theory. The argument is really a balancing act: the component of the acceleration due to the constant speed motion on the curve balances the acceleration due to gravity from the other masses. It is also a local argument: because of the divergence of the acceleration of a filamentary distribution of mass, we have that the local contribution from nearby masses dominates. This link will take you to a gallery of images of the approximate solutions.
This link will take you the Nature web version of the article (Nature login required).
Here is the introductory paragraph (abstract) of the article:
The determination of the exact trajectories of mutually interacting masses (the n-body problem) is apparently intractable for n >2, when the generic solutions become chaotic. A few special solutions are known, which require the masses to be in certain initial positions; these are known as 'central configurations' (an example is the equilateral triangle formed by the Sun, Jupiter and Trojan asteroids). The configurations are usually found by symmetry arguments. Here I report a generalization of the central-configuration approach which leads to large continuous families of approximate solutions. I consider the uniform motion of equidistributed masses on closed space curves, in the limit when the number of particles tends to infinity. In this situation, the gravitational force on each particle is proportional to the local curvature, and may be calculated using an integral closely related to the Biot-Savart integral. Approximate solutions are possible for certain (constant) values of the particle speed, determined by equating this integral to the mass times the centrifugal acceleration. Most smooth, closed space curves support such approximate solutions, because only the local curvature is involved. Moreover, the theory also holds for sets of closed curves, allowing approximate solutions for knotted and linked configurations.
In the same issue is a nice News and Views article by Don Saari discussing the result: Orbits of all sorts Nature, 395, 3 Sept 1998, 19-20 This link will take you Saari's article (Nature login required).
The result was also discussed in web, newspaper, journal, radio and television reports. These include: The New York Times (Science squints at a future fogged by chaotic uncertainty, by Malcolm W. Browne, 22 September 1998). Science News (Following gravityís loops and knots, by Ivars Peterson 5 September 1998 ). Rings Without Planets? by Dana Mackenzie InSCIght 3 September 1998 (InSCIght is a web service of the journal Science).

Other papers in celestial mechanics:
The classical n-body problem is the study of systems of ideal masses moving under the force of gravity. Our solar system is an example, if we model the Sun and the planets by perfect spheres (and ignore any effects of relativity, which matters most for Mercury). The general n-body problem is intractable for three or more masses, in fact it is generically chaotic. What keeps our solar system orderly is the fact that the Sun's mass is so large compared to that of the planets, so it behaves like a collection of two-mass problems, the Sun plus each planet. Since the general problem is intractable, one approach is to look for special solutions -- solutions which we can understand. Central configurations are special positions of the masses which lead to periodic solutions. If we give a central configuration the proper initial angular kick, the entire system rotates rigidly about the center of mass -- as if the masses where attached to a turntable. One way to think of it is that the centrifugal force is balanced perfectly by the gravity. Moreover, if a central configuration is let go from rest, then it collapses homothetically to the center of mass. There is actually an example of such a configuration in our solar system: the Sun, Jupiter, and the Trojan asteroids form an equilateral triangle which rotates about the center of mass of the system.
On clustering in central configurations Proceedings of the American Mathematical Society, 108, 1990, 801-810 In this paper we show that central configurations can not have clusters of masses. For those savvy to the lingo, we showed that in the configuration space the central configurations are bounded away from a neighborhood of the diagonal. In plainer language, we can perhaps guess that there ought to be some sort of bound on clustering -- if two masses were too close together, then if the system were begun at rest they would collide before the entire system had a chance to collapse. The analysis here gives us one of the few geometric conditions known about central configurations.
Mass distributions in collinear central configurations Celestial Mechanics and Dynamical Astronomy, 51, (1991) 305-317 This paper considers the case of equal masses distributed along a line. Moulton showed, in the early part of this century, that the collinear central configuration of n equal masses is unique. In this paper we bounded the mass distribution between a couple functions of n, and provided some numerical evidence of the distribution.
Contact: email: gbuck@anselm.edu Telephone: 603-641-7002 FAX: 603-641-7116 Mechanical Mail: Department of Mathematics Saint Anselm College Manchester, NH 03102