ON THE NUMBER OF CRITICAL CONFIGURATIONS OF CHARGES ON AN M-TORUS
Joseph Bertram Bronder · UA Campus Repository (The University of Arizona) · 1965
The problem of cnumera;: i.nr: the critical (equilibrium) configuration;; of n-charges constrained to lie on an m-dimensional torus is investigated. This problem is equivalent to the enumeration of the stationary values of a real-valued function defined on an m(n-l)-dimensional torus. The law of mutual repulsion is assuraed to be a decreasing function of the square of the 2m-dimensional distance between the charges. The number of critical configurations for the case of two charges is shown to be 2m. For three charges, the number of critical configurations is between 4m and 6m. By appropri ately choosing the law of mutual repulsion and the weights of the charges, both the upper and lower bounds may be attained. The principal tools used in this investigation are some results of M. Morse's topological theory of critical points. A brief development of these results is included.