Minimal biquadratic energy of 5 particles on 2-sphere

TL;DR

This paper investigates the minimal energy configurations of five particles on a sphere under biquadratic potential functions, extending known solutions for specific potentials to this new class.

Contribution

It provides the first known solutions for the five-particle problem on a sphere with biquadratic interaction potentials.

Findings

Derived minimal energy configurations for n=5 with biquadratic potentials

Extended previous solutions from logarithmic and Coulomb potentials to biquadratic case

Contributed to the understanding of energy minimization on spheres for new potential functions

Abstract

Consider n points on the unit 2-sphere. The potential 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 well known. We focus on the case n=5, which turns out to be difficult. In this case, the following results have been obtained. For n=5, Dragnev, Legg, and Townsend (2002) give a solution of the problem for f(r)=-log r known as Whyte's problem. Hou and Shao (2009) give a rigorous computer-aided solution for f(r)=-r. Schwartz (2010) gives a rigorous computer-aided solution of Thompson's problem. We give a solution for biquadratic potentials.