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Rosette central configurations, degenerate central configurations and bifurcations. (English) Zbl 1175.70013

Summary: In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian \(n\)-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where \(n\) particles of mass \(m_1\) lie at the vertices of a regular \(n\)-gon, \(n\) particles of mass \(m_2\) lie at the vertices of another n-gon concentric with the first, but rotated of an angle \(\pi /n\), and an additional particle of mass \(m_0\) lies at the center of mass of the system. This system admits two mass parameters \(\mu = m_0/m_1\) and \(\epsilon = m_2/m_1\). We show that, as \(\mu\) varies, if \(n > 3\), there is a degenerate central configuration and a bifurcation for every \(\epsilon > 0\), while if \(n = 3\) there is a bifurcation only for some values of \(\epsilon\).

MSC:

70F10 \(n\)-body problems
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[4] Hampton, M. and Moeckel, R.: 2006, ’Finiteness of relative equilibria of the four-body problem’, Inv. Math., to appear. · Zbl 1083.70012
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