The Journal of Geometric Mechanics (JGM)

The geometry and dynamics of interacting rigid bodies and point vortices

Pages: 223 - 266, Volume 1, Issue 2, June 2009      doi:10.3934/jgm.2009.1.223

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Joris Vankerschaver - Control and Dynamical Systems, California Institute of Technology MC 107-81, Pasadena, CA 91125, United States (email)
Eva Kanso - Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, United States (email)
Jerrold E. Marsden - Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, United States (email)

Abstract: 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.

Keywords:  Symplectic reduction, Kaluza-Klein, point vortices, perfect fluids.
Mathematics Subject Classification:  Primary: 53D20; Secondary: 76M60.

Received: November 2008;      Revised: March 2009;      Available Online: July 2009.