Thomson’s charge packing problem on a circle (in 2 dimensions)

Question

The Thomson problem in $D$ dimensions is stated as

Find a set of $N$ points $r_1,\ldots, r_N$ on a $(D-1)$-sphere that minimizes the electrostatic potential of the configuration, i.e.,

$$\min_{\|r_n\|=1,\ \forall n} \sum_{n\ne m}\frac{1}{\left\|r_n-r_m\right\|}$$

The problem is open, and for $D\ge 3$, global minima are known only for very few specific values of the number of points $N$.

For $D=2$ dimensions (points on a circumference), intuition strongly suggests that the global minimum is achieved by the vertices of the regular $N$-gon, i.e.,

$$r_n=\left(\cos \frac{2\pi n}{N}, \sin \frac{2\pi n}{N}\right)$$

However, even for this seemingly simple case I have not been able to formally prove that this is a global minimum. It is clearly a local minimum (the gradient is $0$), and I cannot think of any other configurations that make the gradient $0$, but "I cannot think" is not a proof.

Has this been proved and is there a good reference for this? Or can someone come up with a proof?

Answer 1

The 2D Thomson problem has been solved by Harvey Cohn in 1960 using the then recent Morse theory. For equal charges the only stable solution is indeed the regular polygon. In the introduction to Global Equilibrium Theory of Charges on a Circle he writes the following:

"Surprisingly enough the analogous two-dimensional unstable equilibrium problem is far from trivial and yet can be analyzed completely; but it never seems to have made its way into the literature [6]. Here we would consider $n$ point charges, not necessarily equal, constrained to lie on a circle and acting under a fairly general law of repulsion which permits particles to coincide for "closure".

We find all critical configurations, stable and otherwise. The conditions are then perfect for the application of the Poincare-Morse [3] global theory. Here we calculate s, the number of negative squares of the quadratic form that gives the potential near each critical configuration. These values of s are interrelated by a formula ((4.3) below) of the "Euler characteristic" type.

As intuition suggests, there is only one stable critical configuration (excluding rigid rotation) for every ordering of noncoincident charges (for example the obvious regular n-gon for equal charges). The Poincare-Morse theory reduces to identities among partition "tallies" that are not trivial in nature."