Geometry of N-particle clusters in two-dimensions: Some exact results

We report on a study of a finite system of classical confined particles in two-dimensions in the presence of a uniform magnetic field and interacting via a two-body repulsive potential. We develop a simple analytical method of analysis to obtain ground state energies and configurations. We prove analytically the minimum energy configurations are independent of the nature of two body interaction and the magnetic field. In particular we prove that the first transition from a single shell occurs when the number of particles changes from five to six. These results are exact.