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 function for this system can be written as the sum of a completely integrable part H₀ (related to the motion of three point vortices in the plane) and a non-integrable perturbation H₁. Then it is proved that when the vortex strengths all have the same sign and the ratio of the mean distances among the rings is very small in comparison to the mean radius of the rings, H₁/H₀1. Moreover, it is shown that H₁/H₀ is very small for all time for certain initial positions of the rings under the same assumptions. It is proved that the decomposition of the Hamiltonian and the estimates 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 containing quasiperiodic motions. The existence of periodic solutions is also demonstrated. Several examples are solved numerically to show transitions from quasiperiodic and periodic to chaotic regimes in accordance with the theoretical results.