Hamiltonian description of dipoles on the plane and sphere

Poisson bracket for dipoles of ideal 2D hydrodynamics reduces to Zakharov–Faddeev–Gardner bracket by introducing new variables: coordinates of the vorticity extrema and geodesical distances from the poles to the vorticity lines. The new Hamiltonian equations include the well-known two point-vortex system and a decoupled pair of monopoles as asymptotical cases.