Integrable and chaotic motions of four vortices. I. The case of identical vortices

Abstract

It is shown that the three-vortex problem in two-dimensional hydrodynamics is integrable, whereas the motion of four identical vortices is not. A sequence of canonical transformations is obtained that reduces the N-degree-of-freedom Hamiltonian, which describes the interaction of N identical vortices, to one with N - 2 degrees of freedom. For N = 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions. For N = 4 the reduction procedure leads to an effective Hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators. Resonant interaction terms in this Hamiltonian suggest non-integrable behaviour and this is verified by numerical experiments. Explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible. The relevance of the results obtained to fundamental problems in hydrodynamics, such as the question of integrability of Euler's equation in two dimensions, is discussed. The paper also contains a general exposition of the Hamiltonian and Poisson-bracket formalism for point vortices.