Abstract
We consider, after Albouy–Moeckel, the inverse problem for collinear central configurations: Given a collinear configuration of n bodies, find positive masses which make it central. We give some new estimates concerning the positivity of Albouy–Moeckel Pfaffians: We show that for any homogeneity \(\alpha \) and \(n\le 6\) or \(n\le 10\) and \(\alpha =1\) (computer assisted) the Pfaffians are positive. Moreover, for the inverse problem with positive masses, we show that for any homogeneity and \(n\ge 4\) there are explicit regions of the configuration space without solutions of the inverse problem.
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Notes
In the notation of Albouy and Moeckel (2000), \({\varvec{q}}_j = X_j\), \({\varvec{q}}_0=c\), \(A_j=\sum _{k\ne j} m_k {\varvec{Q}}_{ki}\), so that Eq. (1.2) reads as equation (3) of Albouy and Moeckel (2000) \(\alpha A_j - \lambda ({\varvec{q}}_j-{\varvec{q}}_0) =\varvec{0}\), \(j=1,\ldots , n\), for some constant \(\lambda <0\).
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Ferrario, D.L. Pfaffians and the inverse problem for collinear central configurations. Celest Mech Dyn Astr 132, 32 (2020). https://doi.org/10.1007/s10569-020-09975-3
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DOI: https://doi.org/10.1007/s10569-020-09975-3