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Journal of Nonlinear Science

, Volume 24, Issue 2, pp 359–382 | Cite as

Hamiltonian Dynamics of Several Rigid Bodies Interacting with Point Vortices

  • Steffen WeißmannEmail author
Article
  • 170 Downloads

Abstract

We derive the dynamics of several rigid bodies of arbitrary shape in a two-dimensional inviscid and incompressible fluid, whose vorticity is given by point vortices. We adopt the idea of Vankerschaver et al. (J. Geom. Mech. 1(2): 223–226, 2009) to derive the Hamiltonian formulation via symplectic reduction from a canonical Hamiltonian system. The reduced system is described by a noncanonical symplectic form, which has previously been derived for a single circular disk using heavy differential-geometric machinery in an infinite-dimensional setting. In contrast, our derivation makes use of the fact that the dynamics of the fluid, and thus the point vortex dynamics, is determined from first principles. Using this knowledge we can directly determine the dynamics on the reduced, finite-dimensional phase space, using only classical mechanics. Furthermore, our approach easily handles several bodies of arbitrary shape. From the Hamiltonian description we derive a Lagrangian formulation, which enables the system for variational time integrators. We briefly describe how to implement such a numerical scheme and simulate different configurations for validation.

Keywords

Point vortices Fluid-structure interaction Variational integrator Cotangent bundle reduction 

Mathematics Subject Classification (2010)

76B47 53D20 

Notes

Acknowledgments

Ulrich Pinkall proposed the basic idea for deriving the symplectic form. It is my great pleasure to thank him for invaluable discussions and suggestions. Felix Knöppel and David Chubelaschwili helped to work out many of the details. Eva Kanso and the anonymous reviewers provided important feedback for improving the exposition. This work is supported by the DFG Research Center Matheon and the SFB/TR 109 “Discretization in Geometry and Dynamics.”

Supplementary material

Supplementary material 1 (mp4 11302 KB)

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für MathematikTU BerlinBerlinGermany

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