Equilibrium of Sets of Particles on a Sphere

Abstract

A number of particles are constrained to move freely on the surface of a sphere. Between every pair of them there acts a force which depends only on the distance between them, the law of force being the same for all particles. For a given law of force and a given number of particles, there are a number of configurations of the particles in which they are in equilibrium. In general, these configurations will depend on the law of force: we do not expect the particles to remain in equilibrium if the law of force is changed. But there are certain configurations in which the particles are in equilibrium whatever the law of force may be, for instance if the particles are at the vertices of a regular polyhedron. In this paper these configurations are investigated. It is evident that if a configuration has rotational symmetry about the diameter through a certain particle P of it, i.e. if the configuration can be rotated through a fraction of a revolution about the diameter through P and thereby brought into coincidence with its initial position, then the particle P is in equilibrium whatever the law of force may be. Thus among the configurations with the desired property are those which have this rotational symmetry about every diameter through particles of the system. It is shown that these are in fact the only such configurations. Specifically, the only such configurations are those in which the particles are equally spaced on a great circle, with or without two further particles one at each pole, and those which, in relation to a regular polyhedron, comprise a set of particles at its vertices or a set at the mid-points of its edges or a set at the centres of its faces or any two or all three of these sets taken together. (This classification is redundant but it seems the most convenient.) No discussion of the stability of the configurations is given, as this depends in any given case on the law of force, the present discussion being confined to the property of equilibrium with independence of the law of force. The main proof occupies §§ 2-4. In §5 there is discussed the problem of two or more sets of particles on the sphere which are to be in equilibrium whatever the laws of force may be between particles both of the same set or of different sets. §6 relates the present problem to that of maximising the least distance between any two of an assigned number of particles.