New Periodic Solutions for Newtonian $n$-Body Problems with Dihedral Group Symmetry and Topological Constraints

TL;DR

This paper proves the existence of new collision-free periodic solutions in the Newtonian n-body problem with dihedral symmetry, where particles form two twisted regular polygons, using variational methods.

Contribution

It introduces a novel family of symmetric, collision-free periodic solutions for the n-body problem with dihedral group symmetry, expanding known solution classes.

Findings

Existence of new non-collision periodic solutions proven

Solutions exhibit dihedral symmetry with particles forming twisted polygons

Variational minimization and deformation techniques ensure collision-free trajectories

Abstract

In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian $n$-body problems. In our assumption, the $n=2l\geq4$ particles are invariant under the dihedral rotation group $D_l$ in $\mathbb{R}^3$ such that, at each instant, the $n$ particles form two twisted $l$-regular polygons. Our approach is variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.