The symmetric central configurations of the 4-body problem with masses m1=m2≠m

Abstract

We characterize the planar central configurations of the 4body problem with masses m 1 = m 2 ̸ = m 3 = m 4 which have an axis of symmetry. It is known that this problem has exactly two classes of convex central configurations, one with the shape of a rhombus and the other with the shape of an isosceles trapezoid. We show that this 4-body problem also has exactly two classes of concave central configurations with the shape of a kite, this proof is assisted by computer.

Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

“…The restriction to four masses moving on a plane is known as a kite central configuration. The case with three equal masses has been tackled in [7,21] and the case of two couples of equal masses in [4]. Both cases, in the more general situation of charged masses, have been considered in [3].…”

Hjelmslev quadrilateral central configurations

“…Alvarez-Ramírez and Llibre [7] characterized the convex and concave central configurations with an axis of symmetry of the four-body problem when the masses satisfy that m 1 = m 2 = m 3 = m 4 . On the other hand,Érdi and Czirják [17] derived a complete solution in a symmetric case of the planar four-body central configurations, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry.…”

“…and since the masses are positive we get the so called Dziobeck relation(6) (r −3 12 − c 1 )(r −3 34 − c 1 ) = (r −3 13 − c 1 )(r −3 24 − c 1 ) = (r −3 14 − c 1 )(r −3 23 − c 1 ),which holds for every planar central configuration of the 4-body problem. Solving with respect to c 1 any two of these equations we have(7) …”

Equilic quadrilateral central configurations

“…Álvarez-Ramírez and Llibre [7] discussed the convex and concave central configurations with an axis of symmetry of the 4-body problem when the masses satisfy that m 1 = m 2 = m 3 = m 4 . In the same vein, Érdi and Czirják [18] derived a complete solution in a symmetric case of the planar four-body central configurations, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry.…”