A note on the relative equilibria bifurcations in the \((2N+1)\)-body problem (Q258117)

In this paper the authors consider the planar Newtonian \((2N+1)\)-body problem, \(N\geq 1\), with \(2N\) bodies of unit mass at the vertices of two concentric regular \(N\)-gons and one body of mass \(m\) at the center. They prove that solutions form an invariant manifold. Under the above assumptions, and by using bifurcation theory, the authors show that bifurcation of relative equilibria occurs within the dynamics on this manifold when the central mass is allowed to vary near some critical point \(m_c(N)\), \(N\geq 3\). Finally, the authors conclude providing the spectral picture of the linearization at the relative equilibria.