Stability of the Rhomboidal Symmetric-Mass Orbit

We study the rhomboidal symmetric-mass 4-body problem in both a two-degree-of-freedom and a four-degree-of-freedom setting. Under suitable changes of variables in both settings, isolated binary collisions at the origin are regularizable. Linear stability analysis is performed in both settings. For the two-degree-of-freedom setting, linear stability is established for a wide interval of mass ratios. A Poincar\'{e} section analysis is also performed, showing stability. In the four-degree-of-freedom setting, linear stability fails except for a very small interval of mass ratios.