Central Configurations and Mutual Differences

Abstract

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 $j=1,\dots ,n$. 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.

Cited by in corpus (1)

• Central configurations, Morse and fixed point indices