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

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.