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

Abstract

We give a computer assisted proof of the full listing of central configuration for $n$-body problem for Newtonian potential on the plane for $n=5,6,7$ with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For $n=8,9,10$ we establish the existence of central configurations without any reflectional symmetry.