Global dynamics of stationary, dihedral, nearly-parallel vortex filaments

TL;DR

This paper provides an analytical study of the global dynamics of dihedral symmetric vortex filaments interacting via a logarithmic potential, using Celestial Mechanics techniques to understand collision and configuration behaviors.

Contribution

It introduces a novel application of McGehee transformation to vortex filament dynamics with dihedral symmetry, revealing detailed global behavior and stationary configurations.

Findings

Regularization of collision via McGehee transformation

Analysis of invariant manifolds and rest-points for l=2

Connection to stationary configurations of vortex filaments in 3D

Abstract

The goal of this paper is to give a detailed analytical description of the global dynamics of N points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group of order 2l. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as "McGehee transformation". After performing this change of coordinates that regularizes the total collision, we study the rest-points of the flow, the invariant manifolds and we derive interesting information about the global dynamics for l=2. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.