Minimal biquadratic energy of 5 particles on 2-sphere

Abstract

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. In this case, the following results have been obtained. 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. Schwartz [4] gives a rigorous computer-aided solution of Thomson's problem. We give a solution for biquadratic potentials.MSC: 52A40, 52C35.

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“…which follows from the stronger inequality (13) in Lemma 5.1. Observe that equality holds in (18) and (19) if and only a ij + a ji = 0 and d ij + d ji = 0 respectively, which is equivalent to p i • a j + p j • a i = 0, q j • d i + q i • d j = 0, a i = 0, and…”

“…(b) Bi-quadratic potential energy: In [18] it was shown that the TBP is the unique up to rotations optimal spherical configuration of five points on S 2 for the bi-quadratic potential h(x i • x j ) = a (x i • x j ) 2 + b (x i • x j ) + c, a > 0, b > 2a.…”

Distributing many points on spheres: Minimal energy and designs

“…Recently, the papers [179] (for the Coulomb case s = 1 and for s = 2) and [131] (for sum of distances, s = −1) provided computer-assisted proofs that the triangular bi-pyramid is optimal, whereas in the logarithmic case a conventional proof was given in [86]. In [194] a bi-quadratic energy functional is considered. Other rigorously proved minimising configurations are rare and are often universally optimal.…”

Moment methods in energy minimization: New bounds for Riesz minimal energy problems

“…In [35] it is conjectured that the configuration consisting of the vertices of the triangular bipyramid is optimal for 0 < s ≤ s * ≈ 15.048, and the configuration consisting of the vertices of the square pyramid (where the latitude of the base depends on the value of s) is optimal for s ≥ s * . Optimality has been proven for s = 1 and s = 2 by using Hessian bounds and essentially enumerating all possibilities [44], and recently, Schwartz [45] extended his result to the entire interval 0 ≤ s ≤ 6 using an observation of Tumanov [48]. In this paper we are interested in finding sharp dual solutions, which can be used (see, for instance, [13]) to generate easily verifiable optimality proofs; see also the discussion at the end of Section 9.…”