Central Configurations and Mutual Differences

TL;DR

This paper explores the mathematical structure of central configurations in the n-body problem, revealing new equations involving mutual differences and cochains, and providing generalizations of classical results.

Contribution

It introduces a novel formulation linking central configurations to cochain spaces and cocycles, offering new insights and generalizations in the study of these configurations.

Findings

Mutual differences of central configurations satisfy a specific projection equation.

Differences of central configurations are critical points of an analogue of the potential function.

The approach yields immediate generalizations of classical facts in the theory.

Abstract

Central configurations are solutions of the equations $\lambda m_j\boldsymbol{q}_j = \frac{\partial U}{\partial \boldsymbol{q}_j}$, where $U$ denotes the potential function and each $\boldsymbol{q}_j$ 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 $\boldsymbol{q}_{ij} = \boldsymbol{q}_i - \boldsymbol{q}_j$ satisfies the equation $-\frac{\lambda}{\alpha} \boldsymbol{q} = P_m(\Psi(\boldsymbol{q}))$, where $P_m$ is the orthogonal projection over the spaces of $1$-cocycles and $\Psi(\boldsymbol{q}) = \frac{\boldsymbol{q}}{|\boldsymbol{q}|^{\alpha+2}}$. It is shown that differences $\boldsymbol{q}_{ij}$ 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.