Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem

In this paper, we study the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem. The Hamiltonian of the differential system is regularized by a Levi–Civita type transformation and an appropriate scaling of time. The initial condition of this orbit is shown to be the infimum of some well-chosen set. This existence proof is direct and surprisingly simple. Further, a careful study shows that this orbit has a symmetry group isomorphic to the dihedral group D 4 . Then Robertsʼ symmetry reduction method is applied to show the linear stability. It turns out that the rhomboidal periodic orbit in the planar equal mass four-body problem is linearly stable.