Abstract
We derive a symplectic reduction of the evolution equations for a system of three point vortices that proceeds in two steps: a reduction to Jacobi coordinates followed by a Lie-Poisson reduction. In a recent paper, we used a different method in the second step, which took two forms depending on a sign that arises during the calculation. We show that the Lie Poisson reduction unifies these into a single form. We then use this form to succinctly and geometrically explain a kind of bifurcation diagram that has appeared in the literature. We enumerate and plot all the global phase-space diagrams that arise as the circulation of the three vortices is varied. We adapt the reduction method to the case of vanishing total circulation, to which the standard Jacobi reduction does not apply.
