Optimal $$N$$ N -Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

Abstract

This paper inquires into the concavity of the map N → v s (N ) from the integers N ≥ 2 into the minimal average standardized Riesz pair-energies v s (N ) of Npoint configurations on the sphere S 2 for various s ∈ R. The standardized Riesz pair-energy of a pair of points on S 2 a chordal distance r apart is V s (r) = s −1 (r −s − 1), s = 0, which becomes V 0 (r) = ln

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

“…For instance, even the absolutely energy-minimizing 5-particle arrangement on S d changes several times when s varies over the real line, apparently: ( ) For s < −2, all minimizers are either triangular or antipodal [Bjo56], independently of d. In Appendix 1 of [NBK14] it is reported that their own computer-assisted results showed that an antipodal arrangement of two point particles at the South and three at the North Pole is the optimizer for s ≤ −2.368335...; at s = −2.368335... a crossover takes place, and for −2.368335... ≤ s ≤ −2 an isosceles triangle on a great circle, with one particle in the North Pole and two particles each in the other two corners, with (numerically) optimized height, is an energy-minimizing arrangement of five point particles. This has yet to be proved rigorously.…”

“…1 An "optimal energy configuration" is an absolute minimizer of the configurational Riesz s-energy. For background reading concerning the quest for optimal Riesz s-energy configurations of N point particles on S 2 and other manifolds, see the survey articles 2 [ErHo97], [SaKu97], [HaSa04], Appendix 1 in [NBK14], the websites [BCM] and [Wom09], and the related article [AtSu03]. 1 The discontinuous jumps of R s (r) at s = 0 cause some artificial difficulties when trying to compare optimal energies at negative, zero, and positive s values.…”

“…Although the absolute Riesz s-energy minimizers with three particles have been correctly stated (though without detailing the proof) 5 in Appendix 1 of [NBK14], to the best of our knowledge the complete list of critical Riesz senergy arrangements of N = 3 particles, and their stability, has not previously been discussed rigorously. In this paper we will supply such a discussion.…”

“…4 And, of course, it is meaningless to ask for a minimum Riesz pair energy configuration when N = 1, since one cannot form a pair with a single particle alone. 5 Incidentally, there is a slip of pen at the pertinent place in Appendix 1 of [NBK14]: "the only equilibrium arrangements" should read "the only stable equilibrium arrangements." if the p-gradient of the total Riesz s-energy points radially at each particle's position p.…”

“…We remark (see Appendix 3 in[NBK14]) that s → V s (r) is a continuous, monotonically increasing function for all r > 0, strictly so if r = 1, and V s (1) ≡ 0 for all s.…”