Brest State Technical University,

267 Moskowskaya Str.,

224017 Brest,

BELARUS

E-mail: prokopenya@brest.by

**On the Stability of Some Exact Symmetric Solutions
in the Newtonian Many-Body Problem**

**Abstract:**

In our previous work [1] 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.

- Grebenikov E.A., Prokopenya A.N. On the existence of a new class of exact solutions in the planar Newtonian many-body problem, In The questions of modeling and analysis in the problems of making decisions, Ed. V.A.Bereznev, Moscow, Computing Center of RAS, 2004, 39-56.