Numeric-symbolic computations in the study of central configurations in the planar newtonian four-body problem

Abstract

The planar central configurations in the newtonian problem of four bodies are studied with the computer algebra system Mathematica. We have shown that in the case of two equal masses there can exist central configurations in the form of isosceles triangle with three bodies being in its vertices and the fourth body being situated in the axis of symmetry inside or outside the triangle. The number of possible configurations in such cases depends on the masses of the bodies and may be equal to ten, six or two. We have provided evidence numerically that there exist one-parametric family of central configurations in the form of antiparallelogram. We have shown also that central configuration may be deformed continuously by means of changing masses of the bodies and found two-parametric family of central configurations in the neighborhood of the square. © Springer-Verlag Berlin Heidelberg 2006.