Central Configurations and Mutual Differences

Central configurations are solutions of the equations $\lambda m_j\mathbf{q}_j = \frac{\partial U}{\partial \mathbf{q}_j}$, where $U$ denotes the potential function and each $\mathbf{q}_j$ is a point in the $d$-dimensional Euclidean space $E\cong \mathbb{R}^d$, for $j=1,\ldots, n$. We show that the vector of the mutual differences $\mathbf{q}_{ij} = \mathbf{q}_i - \mathbf{q}_j$ satisfies the equation $-\frac{\lambda}{\alpha} \mathbf{q} = P_m(\Psi(\mathbf{q}))$, where $P_m$ is the orthogonal projection over the spaces of $1$-cocycles and $\Psi(\mathbf{q}) = \frac{\mathbf{q}}{|\mathbf{q}|^{\alpha+2}}$. It is shown that differences $\mathbf{q}_{ij}$ 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.