Central configurations in planar n-body problem with equal masses for $$n=5,6,7$$

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...

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...

Title: Saari's Conjecture for the Planar Three-body Problem with Equal Masses Running Title: Saari's Conjecture for Three Equal Masses Author: Saari's Conjecture for the Planar Three-body Problem with Equal Masses

In the N-body problem, it is a simple observation that relative equilib-ria (planar solutions for which the mutual distances between the particles remain constant) have constant moment of inertia. In 1970, Don Saari conjectured that the converse was true: if a solution to the N-body problem has constant moment o f inertia, then it must be a relative equilibrium. In this note, we connrm the conj...

Central configurations and relative equilibria in the n–body problem

Planar Central Configuration Estimates in the N-body Problem

For all masses, there are at least n ?2 O 2-orbits of non-collinear planar central conngurations. In particular, this estimate is valid even if the potential function is not a Morse function. If the potential function is a Morse function, then an improved lower bound, on the order of n! ln ? n+1 3 =2, can be given.