# Hamiltonian Dynamics of Several Rigid Bodies Interacting with Point Vortices

## 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)

## References

- Aref, H.: Point vortex dynamics: a classical mathematics playground. J. Math. Phys.
**48**, 065401 (2007)CrossRefMathSciNetGoogle Scholar - Arnold, V.I.: Sur la géométrie différentielle des groupes de lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Annales de l’institut Fourier
**16**(1), 319–361 (1966)CrossRefMathSciNetGoogle Scholar - Borisov, A.V., Mamaev, I.S.: An integrability of the problem on motion of cylinder and vortex in the ideal fluid. Regul. Chaotic Dyn.
**8**, 163–166 (2003)CrossRefzbMATHMathSciNetGoogle Scholar - Borisov, A.V., Mamaev, I.S., Ramodanov, S.M.: Dynamic interaction of point vortices and a two-dimensional cylinder. J. Math. Phys.
**48**(6), 065403 (2007)CrossRefMathSciNetGoogle Scholar - Chorin, A.: Numerical study of slightly viscous flow. J. Fluid Mech.
**57**, 785–796 (1973)CrossRefMathSciNetGoogle Scholar - Helmholtz, H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. Reine Angew. Math.
**55**, 25–55 (1858)CrossRefzbMATHGoogle Scholar - Kirchhoff, G.R.: Über die Bewegung eines Rotationskörpers in einer Flüssigkeit. Reine Angew. Math.
**71**, 237–262 (1870)CrossRefzbMATHGoogle Scholar - Kobilarov, M., Crane, K., Desbrun, M.: Lie group integrators for animation and control of vehicles. ACM Trans. Graph.
**28**(2009)Google Scholar - Lamb, H.: Hydrodynamics. Cambridge University Press, Cambridge (1895)zbMATHGoogle Scholar
- Lin, C.C.: On the motion of vortices in two dimensions—I and II. Proc. Natl. Acad. Sci. USA
**27**, 570–575 (1941)CrossRefzbMATHGoogle Scholar - Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)Google Scholar
- Marsden, J., Misiolek, G., Ortega, J.P.: Hamiltonian Reduction by Stages. Lecture Notes in Mathematics. Springer, Berlin (2007)Google Scholar
- Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer.
**10**, 357–514 (2001)CrossRefzbMATHMathSciNetGoogle Scholar - Milne-Thomson, L.M.: Theoretical Hydrodynamics, 5th edn. MacMillan and Co., Ltd., London (1968)Google Scholar
- Nair, S., Kanso, E.: Hydrodynamically coupled rigid bodies. J. Fluid Mech.
**592**, 393–411 (2007)CrossRefzbMATHMathSciNetGoogle Scholar - Newton, P.K.: The N-Vortex Problem: Analytical Techniques, Vol. 145 of Applied Mathematical Sciences. Springer, Berlin (2001)CrossRefGoogle Scholar
- Rowley, C.W., Marsden, J.E.: Variational integrators for degenerate Lagrangians, with application to point vortices. Proceedings of 41st IEEE Conference on Decision and Control, vol. 2, pp. 1521–1527 (2002)Google Scholar
- Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
- Shashikanth, B.N.: Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and N point vortices: the case of arbitrary smooth cylinder shapes. Regul. Chaotic Dyn.
**10**(1), 1–14 (2005)CrossRefzbMATHMathSciNetGoogle Scholar - Shashikanth, B.N., Marsden, J.E., Burdick, J.W., Kelly, S.D.: The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices. Phys. Fluids
**14**(3), 1214–1227 (2002)CrossRefzbMATHMathSciNetGoogle Scholar - Shashikanth, B.N., Sheshmani, A., Kelly, S.D., Marsden, J.E.: Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape. Theor. Comput. Fluid Dyn.
**22**, 37–64 (2008)CrossRefzbMATHGoogle Scholar - Vankerschaver, J., Leok, M.: A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects. J. Nonlinear Sci.
**24**(1), 1–37 (2013)Google Scholar - Vankerschaver, J., Kanso, E., Marsden, J.E.: The geometry and dynamics of interacting rigid bodies and point vortices. J. Geom. Mech.
**1**(2), 223–266 (2009)CrossRefzbMATHMathSciNetGoogle Scholar - Weißmann, S., Pinkall, U.: Underwater rigid body dynamics. ACM Trans. Graph.
**31**(4) (2012)Google Scholar