Convex Four-Body Central Configurations with Some Equal Masses

Convex Four Body Central Configurations with Some Equal Masses

We prove that there is a unique convex non-collinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such central configuration posses a symmetry line and it is a kite shaped quadrilateral. We also show that there is exactly on...

Classification of four-body central configurations with three equal masses

It is known that a central configuration of the planar four body problem consisting of three particles of equal mass possesses a symmetry if the configuration is convex or is concave with the unequal mass in the interior. We use analytic methods to show that besides the family of equilateral triangle configurations, there are exactly one family of concave and one family of convex central config...

Action Minimizing Orbits in the Parallelogram Four-body Problem with Equal Masses

The symmetric central configurations of the 4-body problem with masses

We characterize the planar central configurations of the 4body problem with masses m1 = m2 ̸= m3 = m4 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 configu...

Flat Central Configurations of Four Planet Motions

Abst rac t . The flat central configurations of four planet motions are investigated with ~¥u's elimination method. ~¥e obtain 12 collinear central configurations and a necessary condition for determining flat but noncollinear central configurations. We also prove that the number of central configurations in planet motions of 4 bodies is finite under the condition that the masses and angular ve...