The constrained planar N-vortex problem: I. Integrability

The Hamiltonian system governing $N$-interacting particles constrained to lie on a closed planar curve are derived. The problem is formulated in detail for the case of logarithmic (point-vortex) interactions. We show that when the curve is circular with radius $ R $, the system is completely integrable for all particle strengths $ \Gamma _ \beta $, with particle $ \Gamma _ \beta $ moving with frequency $ \omega _ \beta = (\Gamma - \Gamma _ \beta )/4 \pi R^2 $, where $ \Gamma = \sum^{N}_{\alpha=1} \Gamma _ \alpha $ is the sum of the strengths of all the particles. When all the particles have equal strength, they move periodically around the circle keeping their relative distances fixed. When not all the strengths are equal, two or more of the particles collide in finite time. The diffusion of a neutral particle (i.e. the problem of 1D mixing) is examined. On a circular curve, a neutral particle moves uniformly with frequency $ \Gamma / 4 \pi R^2 $. When the curve is not perfectly circular, for example when given a sinusoidal perturbation, or when the particles move on concentric circles with different radii, the particle dynamics is considerably more complex, as shown numerically from an examination of power spectra and collision diagrams. Thus, the circular constraint appears to be special in that it induces completely integrable dynamics.