2007
DOI: 10.1063/1.2428272
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Studies of perturbed three vortex dynamics

Abstract: . It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated to completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the … Show more

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Cited by 18 publications
(13 citation statements)
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“…4 See also Ref. 5 for a recent account of Synge's approach. However, the only systematic analysis of the linear stability problem for the collinear relative equilibria, of which the author is aware, is the 1988 paper by Tavantzis and Ting.…”
Section: Introductionmentioning
confidence: 99%
“…4 See also Ref. 5 for a recent account of Synge's approach. However, the only systematic analysis of the linear stability problem for the collinear relative equilibria, of which the author is aware, is the 1988 paper by Tavantzis and Ting.…”
Section: Introductionmentioning
confidence: 99%
“…This expectation is confirmed by numerical examples presented by Knio et al in Section 23-3 at the annual GAMM-2006 meeting at Berlin. Details of the study and numerical examples are reproduced in a recent paper by Blackmore et al [4], which also presents a proof via the KAM theory. Here we present another proof via asymptotics.…”
mentioning
confidence: 99%
“…In the direct method, we obtain a time-dependent system of differential equations for the motion of a general passive fluid particle by assuming that the dynamics of the pair of coaxial rings has already been determined; i.e., we have the solution (s 1 (t), x 1 (t)), (s 2 (t), x 2 (t)) of (17) and (18), which means that we also know θ 1 (t) := θ (s 1 (t)) and θ 2 (t) := θ (s 2 (t)) owing to (15).…”
Section: Direct Formulation Of Kinematicsmentioning
confidence: 98%