Montserrat Corbera and co-authors have published a new paper to classify all planar four–body central configurations where two pairs of the bodies are on parallel lines. Using cartesian coordinates, they show that the set of four–body trapezoid central configurations with positive masses forms a two–dimensional surface where two symmetric families, the rhombus and isosceles trapezoid, are on its boundary. They also prove that, for a given position of the bodies, in some cases a specific order of the masses determines the geometry of the configuration, namely acute or obtuse trapezoid central configuration. Moreover, they also prove the existence of non–symmetric trapezoid central configurations with two pairs of equal masses.

 

Corbera, M., Cors, J., Llibre, J., Pérez-Chavela, E.(2019) Trapezoid central configurations. Applied Mathematics and Computation, vol. 346, p. 127-142.