The Geometry and Dynamics of Interacting Rigid Bodies and Point Vortices

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 relabelling 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.

arXiv comment52 pages, 2 figures

References in corpus (3)

• Motion of a circular cylinder and n point vortices in a perfect fluid
• Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems
• A nonholonomic Moser theorem and optimal transport

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• The Hydrodynamic Chaplygin Sleigh
• Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid
• The Dynamics of a Rigid Body in Potential Flow with Circulation
• Tulczyjew's Triplet for Lie Groups II: Dynamics
• The movement of a solid in an incompressible perfect fluid as a geodesic flow
• The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation
• Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation
• Fluid-structure interaction in the Lagrange-Poincare formalism: the Navier-Stokes and inviscid regimes
• Hamiltonian dynamics of several rigid bodies interacting with point vortices
• Swimming as a limit cycle