On the dihedral n-body problem

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.

Second version. In the first there was a mistake in a proof: some section had been omitted

References in corpus (2)

• On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem
• Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry

Cited by in corpus (6)

• Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem
• New Periodic Solutions for Newtonian $n$-Body Problems with Dihedral Group Symmetry and Topological Constraints
• Global dynamics of stationary, dihedral, nearly-parallel vortex filaments
• Equilibria of the field generated by point charges
• Symmetric Periodic Orbits in Three Sub-Problems of the N-body Problem
• Global dynamics under a weak potential on a sphere