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Leapfrogging vortexrings: Hamiltonian structure, geometric phases and discrete reduction

Shashikanth, Banavara N. and Marsden, Jerrold E. (2003) Leapfrogging vortexrings: Hamiltonian structure, geometric phases and discrete reduction. Fluid Dynamics Research, 33 (4). pp. 333-356. ISSN 0169-5983. doi:10.1016/j.fluiddyn.2003.05.001.

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We present two interesting features of vortex rings in incompressible, Newtonian fluids that involve their Hamiltonian structure. The first feature is for the Hamiltonian model of dynamically interacting thin-cored, coaxial, circular vortex rings described, for example, in the works of Dyson (Philos. Trans. Roy. Soc. London Ser. A 184 (1893) 1041) and Hicks (Proc. Roy. Soc. London Ser. A 102 (1922) 111). For this model, the symplectic reduced space associated with the translational symmetry is constructed. Using this construction, it is shown that for periodic motions on this reduced space, the reconstructed dynamics on the momentum level set can be split into a dynamic phase and a geometric phase. This splitting is done relative to a cotangent bundle connection defined for abelian isotropy symmetry groups. In this setting, the translational motion of leapfrogging vortex pairs is interpreted as the total phase, which has a dynamic and a geometric component. Second, it is shown that if the rings are modeled as coaxial circular filaments, their dynamics and Hamiltonian structure is derivable from a more general Hamiltonian model for N interacting filament rings of arbitrary shape in R^3, where the mutual interaction is governed by the Biot–Savart law for filaments and the self-interaction is determined by the local induction approximation. The derivation is done using the fixed point set for the action of the group of rotations about the axis of symmetry using methods of discrete reduction theory.

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Additional Information:© 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Received 12 August 2002; accepted 2 May 2003. Communicated by E. Knobloch. The authors would like to thank Darryl Holm and Paul Newton for useful discussions and suggestions. BNS would also like to thank Sergey Pekarsky for some preliminary discussions on cotangent bundle connections.
Subject Keywords:Axisymmetric vortex rings; Leapfrogging motions; Geometric mechanics; Hamiltonian systems; Symmetry reduction; Geometric phases
Issue or Number:4
Classification Code:PACS: 11.15.E, 47.15.K, 05.45, 02.20.T;; 47.10.Df
Record Number:CaltechAUTHORS:20100914-111932765
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19943
Deposited By: Ruth Sustaita
Deposited On:15 Sep 2010 20:37
Last Modified:08 Nov 2021 23:56

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