On the Stability of Some Exact Symmetric Solutions in the Newtonian Many-Body Problem
In our previous work  it has been proved the existence of a general class of exact symmetrical solutions in the Newtonian many-body problem. In the barycentric inertial frame of reference these solutions determine similar closed orbits of the bodies being the conic sections. But in a pulsating, non-uniformly rotating coordinate system they are corresponded to the equilibrium positions of the bodies which form central configurations. In the simplest case two couples of bodies, having equal masses, are situated in the vertices of rhombus symmetrically with respect to its center, where the fifth body of arbitrary mass is resting. The main aim of the present paper is to study the stability of such solutions in the Newtonian problem of five bodies. We have shown that rhombus configuration is unstable for any possible values of the bodies' masses. All the calculations are done with the computer algebra system Mathematica.