Central Configurations and Mutual Differences

Central configurations are solutions of the equations (formula presented), where U denotes the potential function and each qj is a point in the d-dimensional Euclidean space E ≅ Rd, for j = 1…, n. We show that the vector of the mutual differences (formula presented) satisfies the equation (formula presented), where Pm is the orthogonal projection over the spaces of 1-cocycles and (formula presented). It is shown that differences qij of central configurations are critical points of an analogue of U, defined on the space of 1-cochains in the Euclidean space E, and restricted to the subspace of 1-cocycles. Some generalizations of well known facts follow almost immediately from this approach.