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.
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