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Invariant tori in perturbed three vortex motion
Author(s) -
Blackmore Denis,
Ting Lu,
Knio Omar
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700829
Subject(s) - vortex , integrable system , torus , hamiltonian system , quasiperiodic function , kolmogorov–arnold–moser theorem , physics , invariant (physics) , classical mechanics , vortex ring , hamiltonian (control theory) , phase space , mathematical physics , mathematics , geometry , quantum mechanics , mechanics , mathematical optimization , condensed matter physics
The motion of three point vortices in an ideal fluid in a plane comprises a Hamiltonian dynamical system – one that is completely integrable, so it exhibits numerous periodic orbits, and quasiperiodic orbits on invariant tori. Certain perturbations of three vortex dynamics, such as three vortex motion in a half‐plane, are also Hamiltonian, but not completely integrable. Yet these perturbed systems may still have periodic trajectories and invariant tori close to those for the unperturbed dynamics. Extending recent work by the authors [4], invariant 2‐tori approximating those for the unperturbed system are located and analyzed using a combination of classical analysis, asymptotics, and Hamiltonian methods. It is shown that the results and approximation methods used are applicable to several perturbations of three vortex dynamics such as three vortices in a half‐plane, the restricted four vortex problem in the plane, and three coaxial vortex rings in 3‐space. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)