Most smooth closed space curves contain approximate solutions of the n-body problem

Abstract

The determination of the exact trajectories of mutually interacting masses (the n-body problem¹,²) is apparently intractable for n ⩾ 3, 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’ (refs 1,2,3,4,5,6) (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 contain 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.