On a discrete variational problem involving interacting particles

We consider the minimization problem [GRAPHICS] for z = (z(1), z(2),..., z(N)) is an element of C-N satisfying Sigma(k=1)(N) \z(k)\(2) = 1 and z(k) not equal z(j) for 1 less than or equal to k < j less than or equal to 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 greater than or equal to 6), and if N greater than or equal to 11 it cannot be a star configuration (i.e., an N + 1 sides regular polygon plus the origin).