Hjelmslev quadrilateral central configurations

Abstract

A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. We classify all planar four-body central configurations where the four bodies are at the vertices of a Hjelmslev quadrilateral. We show that in the positive mass space (m 1 , m 2 , m 3 , m 4), taking the unit of mass equal to m 1 , the set of Hjelmslev quadrilateral central configurations of the fourbody problem is an open arc. When the masses tend to the endpoints of this arc three of the masses of the Hjelmslev quadrilateral central configurations tend to an equilateral central configuration of the three-body problem, and the fourth remainder mass tends to zero.

Restricted Concave Kite Five-Body Problem

“…Tese exceptional cases are given explicitly as polynomials in the masses (or vorticities in the vortex problem). Álvarez-Ramírez and Llibre [15] using mutual distances as coordinates showed that any fourbody central confguration forming a Hjelmslev quadrilateral must be a right kite confguration. A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.…”