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

Francesco Paparella; Alessandro Portaluri

doi:10.3934/dcds.2013.33.3011

Article Contents

2013, Volume 33, Issue 7: 3011-3042. Doi: 10.3934/dcds.2013.33.3011

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

Francesco Paparella^1, and
Alessandro Portaluri^2,

1.
Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, 73100, Lecce
2.
Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, 73100 Lecce

Received: March 31, 2012

Revised: September 30, 2012

Published: June 2013

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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 $D_l$ 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, 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.

Keywords:

Mathematics Subject Classification: Primary: 70F10; Secondary: 37C80, 76B47.

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