```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?
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