Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 021, 11 pages      arXiv:1608.00480      https://doi.org/10.3842/SIGMA.2017.021

Central Configurations and Mutual Differences

D.L. Ferrario
Department of Mathematics and Applications, University of Milano-Bicocca, Via R. Cozzi, 55 20125 Milano, Italy

Received December 06, 2016, in final form March 27, 2017; Published online March 31, 2017

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.

Key words: central configurations; relative equilibria; $n$-body problem.

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