The Dynamics of a Rigid Body in Potential Flow with Circulation

Details The Dynamics of a Rigid Body in Potential Flow with Circulation The Dynamics of a Rigid Body in Potential Flow with Circulation

2010-02-27

arxiv.org/abs/1003.0080v2

Regular Chaot.Dyn.15:606-629,2010

Physics

Mathematical Physics

28 pages, 2 figures; v2: typos corrected

Scientific paper

10.1134/S1560354710040143

We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor.

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