  We study equilibrium configurations of infinitely many identical particles on
the real line or finitely many particles on the circle, such that the
(repelling) force they exert on each other depends only on their distance. The
main question is whether each equilibrium configuration needs to be an
arithmetic progression. Under very broad assumptions on the force we show this
for the particles on the circle. In the case of infinitely many particles on
the line we show the same result under the assumption that the maximal (or the
minimal) gap between successive points is finite (positive) and assumed at some
pair of successive points. Under the assumption of analyticity for the force
field (e.g., the Coulomb force) we deduce some extra rigidity for the
configuration: knowing an equilibrium configuration of points in a half-line
determines it throughout. Various properties of the equlibrium configuration
are proved.
