Date: Tue, 26 Mar 96 13:33:07 CST
From: rusin (Dave Rusin)
To: ksbrown@ksbrown.seanet.com
Subject: Re: Most Wanted List
Newsgroups: sci.math

In article <4j48aj$ja6@kaleka.seanet.com> you write:
>(10) Let c(N) denote the number of distinct configurations of 
>     N equally charged particles in static equilibrium on the 
>     surface of a sphere of radius R.  How does c(N) vary (if at
>     all) with R?

My geometric intuition suggests that the equilibrium configurations
vary continuously in configuration space X=(S^2)^N  with  R. The
phrasing of your question suggests that you have a counterexample to
the expected stronger result: if as  R  varies, one point is held
fixed and the great circle joining it and another given point is also
held constant, then the equilibrium configurations trace out
c(N) disjoint paths in  X  as  R  varies over  (0, oo). Is there
some phenomenon you know of which prohibits this stronger version?
(e.g. an example of two configurations which coalesce at some  R,
or one which bifurcates as  R  increases, or one which cannot be
continued past some min. or max. value of  R)?

(It may be easier to view the problem this way: rather than varying
the radius  R, keep the points on a sphere of radius  1, but vary
the constant of proportionality between the force and the
(inverse of the square of the) distance. Taking the radius to infinity
corresponds to taking the force to zero, making the problem purely
local and thus forcing an approximation of the kissing pennies at
each point; taking the radius to  zero  corresponds to taking the
force to infinity, making the problem really global; I don't know
what this tends to make the solutions look like.)

dave