# American Institute of Mathematical Sciences

July  2013, 33(7): 3011-3042. doi: 10.3934/dcds.2013.33.3011

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

 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  April 2012 Revised  October 2012 Published  January 2013

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.
Citation: Francesco Paparella, Alessandro Portaluri. Geometry of stationary solutions for a system of vortex filaments: A dynamical approach. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3011-3042. doi: 10.3934/dcds.2013.33.3011
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