Central Configurations and Mutual Differences

Abstract

Abstract. Central configurations are solutions of the equations λm j q j = ∂U ∂qj , where U denotes the potential function and each q j is a point in the d-dimensional Euclidean space E ∼ = R d , for j = 1, . . . , n. We show that the vector of the mutual differences q ij = q i − q j satisfies the equation − λ α q = P m (Ψ(q)), where P m is the orthogonal projection over the spaces of 1-cocycles and Ψ(q) = q |q| α+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.

Pfaffians and the inverse problem for collinear central configurations

“…A central configuration is a configuration that yields a relative equilibrium solution of the Newton equations of the n-body problem with potential function U , and can be shown (cf. [13], [12], [1], [6], [8], [7]) that it is a solution of the following n equations (1.1) λm j q j = −α k =j m j m k q j − q k q j − q k α+2 .…”

Central configurations, Morse and fixed point indices

“…[17] ( §369- §382bis at pp. 284-306), [15], [10], [12], [18], [1], [6], [2], [11], [5]). An equivalent definition for a normalized (i.e.…”