Problem 10 on my Most Wanted list has evolved somewhat since the
list was first posted.  Several years ago I made a fairly detailed 
study of the equilibrium arrangements of points on a sphere with 
inverse-square repulsion, and noticed the embedding occurrances that 
led to what became Problem 11 on my List.  Of course, one of the first 
things one realizes about arrangements of points on a sphere with 
inverse-square repulsion is that the size of the sphere is 
unimportant, because the potential scales proportionately.  However, 
as I was typing Problem 11 I began to wonder about the effects of 
scale on sets of repulsive particles.  I had lost sight of the
scale invariance of inverse-power laws, and it seemed like an 
interesting question at the time, so I rashly entered it as 
Problem 10.  

Well, John Samson immediately noticed my screw-up and sent the 
following nice explanation of scale-invariance for inverse-power 
laws:

> Assuming the charges interact via the Coulomb interaction, (and 
> that it's a classical problem) there is no length scale.  The 
> potential energy is
>
>         V({r_i}) = K \sum_i \sum_j < i   1/|r_i-r_j|
>
> where K is a constant, Q^2/(4pi epsilon_0) if you like, and
>
>                  r_i = R e_i
>
> are the position vectors of the charges, where e_i are unit
> vectors.  Then clearly the potential scales as 1/R
>
>               V({r_i}) = V({e_i})/R
>
> and the extrema (and number of extrema up to symmetries) will be
> independent of R.   The same applies for any power-law force; the 
> potential energy function will scale as r^(-k+1).  The answer is 
> the same even if the charges are allowed to enter the interior, 
> |r_i| <= R.
>
> Now suppose there is a length scale - say a Yukawa rather than a
> Coulomb potential.  The problem is left as an exercise...
>
> -------------------------------------------------------------------
> John Samson, Department of Physics,        Lost Consonants 1:
> Loughborough University of Technology      I fear geeks bearing .gifs
> Home page  http://www.lboro.ac.uk/departments/ph/phjhs/
> -------------------------------------------------------------------


By the way, the Yukawa potential mentioned by John is

                  V(r) ~ (1/r)*exp(-r/r0)

compared with the generalized Coulomb potential

                  V(r) ~ (1/r)*(r0/r)^(k-2)

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 so fast with increasing separation 
that I wondered if it necessarily has more than one equilibrium 
configuration for any given N and R.  John Samson replied:

> With the Yukawa potential, for any fixed N, you'd presumably get 
> the same result as for the Coulomb potential for R/r0 less than 
> some epsilon.  Once the ratio exceeds this, the critical points 
> of the energy landscape might merge or bifurcate and c(N) might 
> change.

Then I asked if there is any known potential function that gives 
a unique equilibrium configuration for each N and R.  John commented

> I'd go for the answer "no" for isotropic two-body repulsive 
> interactions, but wouldn't know how to prove it.  You are asking 
> for a (2N-2)-dimensional energy landscape to have only one minimum 
> for any N, which sounds hard to arrange.  If the interaction is 
> attractive, there clearly is a unique minimum.  If you allow 
> anisotropic or three-body interactions, you could probably arrange 
> for the charges to lie on a great circle.

John's comments about force laws that possess a definite scale
prompted me to suggest 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 = -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 you 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 is raised by John's remark that 
"the critical points of the energy landscape might merge or 
bifurcate and c(N) might change".  Is merging and bifurcating 
of critical points the only ways for c(N) to change with R?
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?  Hmmm...

Anyway, as a result of the above discussion I modified Problem 10
on my Most Wanted List to refer to a Cauchy force law.  It still
seems like an interesting question to me.