Equilibria and stability in the restricted \((n+1)\)-body problem with logarithm potential

Abstract

We study the existence and stability of equilibria in the regular n-gon restricted \((n+1)\)-body problem with logarithm potential. We determine two classes of equilibria: “infinitesimal-Eulerian” situated along lines joining the n-gon centre with a vertex (i.e. along the radii), and “infinitesimal-Lagrangian” situated on the perpendicular bisectors of the n-gon sides. The infinitesimal-Eulerian equilibria are all positioned outside the primaries n-gon and are unstable. The infinitesimal-Lagrangian equilibria appear in two families: an unstable family in the interior of the primaries’ polygon, and a linearly stable family in the exterior. We also prove the existence of an equilibrium at the centre of the polygon that is unstable.