Geometric properties of minimizers in the planar three-body problem with two equal masses

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...

Shape Space Figure 8 Solutions of Three Body Problem with Two Equal Masses

In an unpublished paper by Montgomery [18], the author claimed to prove the existence of shape space Figure-8 solutions of the Newtonian three body problem with two equal masses (they look like a figure 8’s in the shape space, that is different from the famous Figure-8 solution proved in [11]). However there is an error in the proof and the problem is still open. In this paper, based on a new l...

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...

A remarkable periodic solution of the three - body problem in the case of equal masses

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in ...

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