KAM theory analysis of the dynamics of three coaxial vortex rings

The equations of motion of three coaxial vortex rings in Euclidean 3-space are formulated as a Hamiltonian system. It is shown that the Hamiltonian fu nction for this system can be written as the sum of a completely integrable part H-0 (related to the motion of three point vortices in the plane) and a non-integrable perturbation H-1. Then it is proved that when the vortex s trengths all have the same sign and the ratio of the mean distances among t he rings is very small in comparison to the mean radius of the rings, H-1/H -0 much less than 1. Moreover, it is shown that H-1/H-0 is very small for a ll time for certain initial positions of the rings under the same assumptio ns. It is proved that the decomposition of the Hamiltonian and the estimate s carry over to a reduced form of the system in coordinates moving with the center of vorticity and having one less degree of freedom. Then KAM theory is applied to prove the existence of invariant two-dimensional tori contai ning quasiperiodic motions. The existence of periodic solutions is also dem onstrated, Several examples are solved numerically to show transitions from quasiperiodic and periodic to chaotic regimes in accordance with the theor etical results. (C) 2000 Elsevier Science B.V. All rights reserved.