The three-vortex system: Hopf fibration, symplectic reduction, and near-collisions

Abstract: We study the dynamics of collisions and near-collisions in the system of three point vortices in the plane. The dynamics is known to be completely integrable and can be reduced to the study of Hamiltonian systems on two-dimensional symplectic leaves. This still allows for interesting behavior of the solutions, various aspects of which have been investigated in a large body of work by many authors. Our focus will be on collisions and the possibility of their regularization through perturbations of suitable parameters of the system. We will use the reduced coordinate given by $ζ=(z_2-z_1)/(z_3-z_1)$, where $z_1,z_2,z_3$ are the complex numbers representing the positions of the vortices. This choice seems very natural, although we have not come across it in the existing literature on the topic. Some of our results concerning the near-collision behavior appear to be new, and we also prove inequalities between critical energy values that we have not found in the literature. The calculations we do to analyze the collisions and near-collisions also reproduce various classical results that were previously obtained by other authors using different coordinates, and parts of the paper therefore have an expository character.