Integrable and chaotic motions of four vortices. I  The case of identical vortices
Abstract
It is shown that the threevortex problem in twodimensional hydrodynamics is integrable, whereas the motion of four identical vortices is not. A sequence of canonical transformations is obtained that reduces the Ndegreeoffreedom 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 nonintegrable 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 Poissonbracket formalism for point vortices.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 April 1982
 DOI:
 10.1098/rspa.1982.0047
 Bibcode:
 1982RSPSA.380..359A
 Keywords:

 Computational Fluid Dynamics;
 Equations Of Motion;
 Ocean Dynamics;
 Stochastic Processes;
 Turbulence;
 Vortices;
 Canonical Forms;
 Degrees Of Freedom;
 Elliptic Functions;
 Hamiltonian Functions;
 Hydrodynamic Equations;
 Fluid Mechanics and Heat Transfer