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 D4. 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.