Central Configurations and Mutual Differences

Central configurations are solutions of the equations lambda m(j)q(j) = partial derivative U/partial derivative q(j), where U denotes the potential function and each q(j) is a point in the d-dimensional Euclidean space E congruent to R-d, for j = 1, . . . , n. We show that the vector of the mutual differences q(ij) = q(i) - q(j) satisfies the equation -lambda/alpha q = P-m (Psi(q)), where P-m is the orthogonal projection over the spaces of 1-cocycles and Psi (q) = q/vertical bar q vertical bar (alpha+2). It is shown that differences q(ij) 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.