Geometry of stationary solutions for a system of vortex filaments: A dynamical approach

Francesco Paparella, Alessandro Portaluri

Research output: Contribution to journalArticle

Abstract

We 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 Dl 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 ow, the invariant manifolds and, with the help of a computer algebra system, 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.

Original languageEnglish (US)
Pages (from-to)3011-3042
Number of pages32
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume33
Issue number7
DOIs
StatePublished - Jul 1 2013

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Keywords

  • Dihedral N-vortex filaments
  • Global dynamics
  • Logarithmic potential
  • McGehee coordinates
  • N-body problems

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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