| One is unlikely to overestimate the importance or difficulty of the Newtonian gravitational n-body problem. The general problem appears to be intractable for three or more bodies; it is generically chaotic. There is a long history of investigations of special solutions. Few special solutions are known, those that are require initial configurations of the masses which are usually found through arguments using symmetry. In this work we take another approach. We analyse the behavior of masses distributed along sets of closed space curves. This analysis leads to a novel result: large (continuous) families of approximate solutions. In these approximate solutions 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, and involves the analysis of an integral of the same form as the Biot-Savart integral |
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