Minimal biquadratic energy of five particles on a 2-sphere

Consider n points on the unit 2-sphere. The potential energy of the interaction of two points is a function f (r ) of the distance r between the points. The total energy E of n points is the sum of the pairwise energies. The question is how to place the points on the sphere to minimize the energy E. For the Coulomb potential f (r ) = 1/r, the problem goes back to Thomson (1904). The results for n < 5 are simple and well known; we focus on the case n = 5, which turns out to be difficult. Dragnev, Legg, and Townsend [2] give a solution of the problem for f (r ) = -log r (known as Whyte's problem). Hou and Shao give a rigorous computer-aided solution for f (r ) = -r, while Schwartz [4] gives one for Thomson's problem. Finally, we give a solution for biquadratic potentials.