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The geometry and dynamics of interacting rigid bodies and point vortices

Vankerschaver, Joris and Kanso, Eva and Marsden, Jerrold (2009) The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 1 (2). pp. 223-266. ISSN 1941-4889.

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We derive the equations of motion for a planar rigid body of circular shape moving in a 2D perfect fluid with point vortices using symplectic reduction by stages. After formulating the theory as a mechanical system on a configuration space which is the product of a space of embeddings and the special Euclidian group in two dimensions, we divide out by the particle relabeling symmetry and then by the residual rotational and translational symmetry. The result of the first stage reduction is that the system is described by a non-standard magnetic symplectic form encoding the effects of the fluid, while at the second stage, a careful analysis of the momentum map shows the existence of two equivalent Poisson structures for this problem. For the solid-fluid system, we hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden, Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the other hand. As a side result, we obtain a convenient expression for the symplectic leaves of the reduced system and we shed further light on the interplay between curvatures and cocycles in the description of the dynamics.

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Additional Information:© AIMS 2009. Received: November 2008; revised: March 2009; published: July 2009. It is a pleasure to thank Jair Koiller, Richard Montgomery, Paul Newton, Tudor Ratiu and Banavara Shashikanth, as well as Frans Cantrijn, Scott Kelly, Bavo Langerock, Jim Radford, and Clancy Rowley, for useful suggestions and interesting discussions. J. Vankerschaver is a Postdoctoral Fellow from the Research Foundation – Flanders (FWO-Vlaanderen) and a Fulbright Research Scholar, and wishes to thank both agencies for their support. Additional financial support from the Fonds Professor Wuytack is gratefully acknowledged. E. Kanso’s work is partially supported by NSF through the award CMMI 06-44925. J. E. Marsden is partially supported by NSF Grant DMS-0505711.
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Fubright ProgramUNSPECIFIED
Fonds Professor WuytackUNSPECIFIED
NSFCMMI 06-44925
Subject Keywords:Symplectic reduction; Kaluza-Klein; point vortices; perfect fluids
Classification Code:Mathematics Subject Classification: Primary: 53D20; Secondary: 76M60.
Record Number:CaltechAUTHORS:20091102-123126604
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16540
Deposited By: Jason Perez
Deposited On:03 Nov 2009 22:36
Last Modified:26 Dec 2012 11:31

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