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Regular and Chaotic Dynamics

, Volume 15, Issue 4–5, pp 606–629 | Cite as

The dynamics of a rigid body in potential flow with circulation

  • J. VankerschaverEmail author
  • E. Kanso
  • J. E. Marsden
Special Issue: Valery Vasilievich Kozlov-60

Abstract

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.

Key words

fluid-structure interactions potential flow circulation symplectic reduction diffeomorphism groups oscillator group 

MSC2000 numbers

76B47 53D20 74F10 

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Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Control and Dynamical SystemsCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Dept. of Mathematical Physics and AstronomyGhent UniversityGhentBelgium
  3. 3.Aerospace and Mechanical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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