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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.
Mathematics Subject Classification: Primary: 53D20; Secondary: 76M60.
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