When considering the stable configurations of n point-like particles on the surface of a sphere under the assumption that the particles repel each other with an inverse-square force, it's not hard to see that the size of the sphere is unimportant, because the potential scales proportionately. In other words, noting that the total potential energy of the system is P = SUM 1/|ri - rj| where the sum is taken over all distinct pairs i,j, it's clear that if we re-scale all lengths by a factor k the new potential is P/k. This constant re-scaling doesn't affect any of the maxima or minima, so the equilibrium configurations are unaffected. Obviously the same applies to any pure power law, such as an inverse-cube repulsion. We could also consider potentials with a definite scale, such as the generalized Coulomb potential V(r) ~ (1/r)*(r0/r)^(k-2) or the Yukawa potential V(r) ~ (1/r)*exp(-r/r0) The Yukawa approximates certain nucleon interactions, whereas I'm not aware of any physical applications of generalized Coulomb potential (aside form k=2). The Yukawa potential drops off very rapidly with increasing separation, so it would be interesting to know how the asymptotic number of distinct equilibrium configurations differs from that for power laws. Also, is there any known potential function that gives a unique equilibrium configuration for each N and R? It seems unlikely, but not easy to prove. Speaking of force laws that possess a definite scales, we could also consider a "Cauchy" force law of the form F ~ 1/(1+r^2), which could easily be mistaken for an inverse square law at long range. I think the corresponding potential function is something like V(r) = -invtan(r) Unlike the power laws, the scale can affect the ratio of the potentials associated with two different separations, as shown by the fact that invtan(3)/invtan(2) is not equal to invtan(6)/invtan(4). If we substitute 1/(1+r^2) (with some specific choice of scale factor for r) in place of Newton's 1/r^2, would the solution of the two-body problem still be a stable ellipse? Or might it be a precessing ellipse? Hmmm... Another interesting question concerns how the local minima might change as the scale factor is continuously varied. The number of minimal configurations need not be conserved, because the critical points could merge or bifurcate. Is merging and bifurcating of critical points the only ways for the number of minimal configurations to change with the scale factor? Would it be possible for a region containing a local minimum to get flatter and flatter, and at some R change from being concave to convex, so the local minimum simply dissappears, rather than merging into another one?