On a discrete variational problem involving interacting particles

We consider the minimization problem min F(z)≡-qqlog|z_k-z_j|, k,j = 1, 2, ..., N, for z = (z₁, z₂, ..., z_N)∈C^N satisfying Σ_{k = 1}^N|z_k|² = 1 and z_k≠z_j for 1≤k<j≤N. This problem arises in different contexts in several physical models such as incompressible Euler equations and the Ginzburg-Landau model in superconductivity. We study the stability properties of some symmetric critical points such as regular polygons and configurations that consist of a regular polygon plus the origin. We also establish the existence of an infinite number of critical points enjoying different kinds of symmetry. We show that when the number of particles, N, exceeds a critical value, the global minimizer cannot be a regular polygon (N≥6), and if N≥11 it cannot be a star configuration (i.e., an N-1 sides regular polygon plus the origin).