On the dihedral n-body problem

TL;DR

This paper studies a symmetric n-body problem with dihedral symmetry, finding all central configurations and analyzing the stability of their manifolds, extending classical problems with new symmetry constraints.

Contribution

It introduces a novel symmetric n-body problem with dihedral symmetry and explicitly characterizes all central configurations and their stability properties.

Findings

All central configurations are classified.

Dimensions of stable and unstable manifolds are computed.

The problem generalizes classical symmetric n-body problems.

Abstract

Consider n=2l>=4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group D_l, where D_l is the group of order 2l generated by two rotations of angle pi around two secant lines in space meeting at an angle of pi/l. By adding a homogeneous gravitational (Newtonian) potential one finds a special $n$-body problem with three degrees of freedom, which is a kind of generalisation of Devaney isosceles problem, in which all orbits have zero angular momentum. In the paper we find all the central configurations and we compute the dimension of the stable/unstable manifolds.