Thomson Problem

Determine the stable equilibrium positions of $N$ classical electrons constrained to move on the surface of a Sphere and repelling each other by an inverse square law. Exact solutions for $N=2$ to 8 are known, but $N=9$ and 11 are still unknown.

In reality, Earnshaw's theorem guarantees that no system of discrete electric charges can be held in stable equilibrium under the influence of their electrical interaction alone (Aspden 1987).

See also Fejes Tóth's Problem

References

Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Dowla, F.; and Wooten, F. ``Method of Constrained Global Optimization.'' Phys. Rev. Let. 72, 2671-2674, 1994.

Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Dowla, F.; and Wooten, F. ``Method of Constrained Global Optimization--Reply.'' Phys. Rev. Let. 74, 1483, 1995.

Ashby, N. and Brittin, W. E. ``Thomson's Problem.'' Amer. J. Phys. 54, 776-777, 1986.

Aspden, H. ``Earnshaw's Theorem.'' Amer. J. Phys. 55, 199-200, 1987.

Berezin, A. A. ``Spontaneous Symmetry Breaking in Classical Systems.'' Amer. J. Phys. 53, 1037, 1985.

Calkin, M. G.; Kiang, D.; and Tindall, D. A. ``Minimum Energy Configurations.'' Nature 319, 454, 1986.

Erber, T. and Hockney, G. M. ``Comment on `Method of Constrained Global Optimization.''' Phys. Rev. Let. 74, 1482-1483, 1995.

Marx, E. ``Five Charges on a Sphere.'' J. Franklin Inst. 290, 71-74, Jul. 1970.

Melnyk, T. W.; Knop, O.; and Smith, W. R. ``Extremal Arrangements of Points and Unit Charges on a Sphere: Equilibrium Configurations Revisited.'' Canad. J. Chem. 55, 1745-1761, 1977.

Whyte, L. L. ``Unique Arrangement of Points on a Sphere.'' Amer. Math. Monthly 59, 606-611, 1952.