Point vortex equilibria and optimal packings of circles on a sphere

摘要

We answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of N =3-12 and 24, the only values of N for which the optimal packing problem (also known as the Tammes problem) has rigorously known solutions. We also address the cases N =13-23 where optimal packing solutions have been computed, but not proven analytically. As in Jamaloodeen & Newton (Jamaloodeen & Newton 2006 Proc. R. Soc. Lond. Ser. A 462, 3277-3299. (doi:10.1098/rspa.2006.1731)), a logarithmic, or point vortex equilibrium is determined by formulating the problem as the one in linear algebra, A→Γ =0, where A is a N(N - 1)/2 × N non-normal configuration matrix obtained by requiring that all interparticle distances remain constant. If A has a kernel, the strength vector →Γ ?N is then determined as a right-singular vector associated with the zero singular value, or a vector that lies in the nullspace of A where the kernel is multi-dimensional. First we determine if the known optimal packing solution for a given value of N has a configuration matrix A with a non-empty nullspace. The answer is yes for N =3-9, 11-14, 16 and no for N =10, 15, 17-24. We then determine a basis set for the nullspace of A associated with the optimally packed state, answer the question of whether N-equal strength particles, →Γ =(1, 1, 1, ?, 1)T, form an equilibrium for this configuration, and describe what is special about the icosahedral configuration from this point of view. We also find new equilibria by implementing two versions of a random walk algorithm. First, we cluster sub-groups of particles into patterns during the packing process, and grow a packed state using a version of the yin-yang algorithm of Longuet-Higgins (Longuet-Higgins 2008 Proc. R. Soc. A (doi:10.1098/rspa.2008.0219)). We also implement a version of our Brownian ratchet algorithm to find new equilibria near the optimally packed state for N =10, 15, 17-24.