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


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, R., Marsden, J.E., and Ratiu, T., Manifolds, Tensor Analysis, and Applications. Second Edition. Applied Mathematical Sciences, vol. 75, New York: Springer, 1988.zbMATHGoogle Scholar
  2. 2.
    Abraham, R. and Marsden, J.E. Foundations of Mechanics. Second edition, revised and enlarged. With the assistance of Tudor Ratiu and Richard Cushman. Reading, Mass.: Benjamin/Cummings Publishing Co., 1978.zbMATHGoogle Scholar
  3. 3.
    Arnold, V., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 1966, vol. 16, pp. 319–361.Google Scholar
  4. 4.
    Arnold, V.I., Kozlov, V.V., and Neishtadt, A.I., Mathematical Aspects of Classical and Celestial Mechanics, Berlin: Springer, 2006.zbMATHGoogle Scholar
  5. 5.
    Arnold, V.I. and Khesin, B.A., Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, New York: Springer, 1998.zbMATHGoogle Scholar
  6. 6.
    Batchelor, G.K., An Introduction to Fluid Dynamics. Second paperback edition. Cambridge Mathematical Library, Cambridge: Cambridge University Press, 1999.zbMATHGoogle Scholar
  7. 7.
    Borisov, A.V., Kozlov, V.V., and Mamaev, I.S., Asymptotic Stability and Associated Problems of Dynamics of Falling Rigid Body, Regul. Chaotic Dyn., 2007, vol. 12, pp. 531–565.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Borisov, A.V. and Mamaev, I.S., On theMotion of a Heavy Rigid Body in an Ideal Fluid with Circulation, Chaos, 2006, vol. 16, 013118, 7 pp.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Borisov, A.V., Mamaev, I.S., and Ramodanov, S.M., Dynamic Interaction of Point Vortices and a Two-dimensional Cylinder, J. Math. Phys., 2007, vol. 48, 065403, 9 pp.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Cendra, H., Marsden, J., and Ratiu, T.S., Cocycles, Compatibility, and Poisson Brackets for Complex Fluids. In Advances in Multifield Theories for Continua with Substructure, Model. Simul. Sci. Eng. Technol., Boston, MA: Birkhäuser Boston, 2004, pp. 51–73.Google Scholar
  11. 11.
    Chaplygin, S.A., On the Effect of a Plane-parallel Air Flow on a Cylindrical Wing Moving in It, The Selected Works on Wing Theory of Sergei A. Chaplygin., 1956, pp. 42–72. Translated from the 1933 Russian original by M.A. Garbell.Google Scholar
  12. 12.
    Ebin, D.G. and Marsden, J.E., Groups of Diffeomorphisms and the Notion of an Incompressible Fluid, Ann. Math. (2), 1970, vol. 92, pp. 102–163.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Gay-Balmaz, F. and Ratiu, T.S., Affine Lie-Poisson Reduction, Yang-Mills Magnetohydrodynamics, and Superfluids, Journal of Physics A: Mathematical and Theoretical, 2008, vol. 41, 344007, 24pp.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Gay-Balmaz, F. and Ratiu, T.S., The Geometric Structure of Complex Fluids, Adv. in Appl. Math., 2009, vol. 42, pp. 176–275.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Guillemin, V. and Sternberg, S., Symplectic Techniques in Physics, Cambridge: Cambridge University Press, 1984.zbMATHGoogle Scholar
  16. 16.
    Holm, D.D. and Kupershmidt, B.A., The Analogy Between Spin Glasses and Yang-Mills Fluids, Journal of Mathematical Physics, 1988, vol. 29, pp. 21–30.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kanso, E., Marsden, J.E., Rowley, C.W., and Melli-Huber, J.B., Locomotion of Articulated Bodies in a Perfect Fluid, J. Nonlinear Sci., 2005, vol. 15, pp. 255–289.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kanso, E. and Oskouei, B., Stability of a Coupled Body-Vortex System, J. Fluid Mech., 2008, vol. 600, pp. 77–94.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kanso, E., Swimming Due to Transverse Shape Deformations, J. Fluid Mech., 2009, vol. 631, pp. 127–148.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry. Vol. 1, New York: Interscience Publishers, John Wiley & Sons. 1963.Google Scholar
  21. 21.
    Koiller, J. Note on Coupled Motions of Vortices and Rigid Bodies, Phys. Lett. A, 1987, vol. 120, pp. 391–395.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Kozlov, V.V., On a Heavy Cylindrical Body Falling in a Fluid, Izv. RAN, Mekh. tv. tela, 1993, no. 4, pp. 113–117.Google Scholar
  23. 23.
    Kozlov, V.V., General Theory of Vortices. Dynamical systems. X, Encyclopaedia of Mathematical Sciences, vol. 67, Berlin: Springer, 2003. Translated from the 1998 Russian edition.Google Scholar
  24. 24.
    Lamb, H., Hydrodynamics, Dover Publications, 1945. Reprint of the 1932 Cambridge University Press edition.Google Scholar
  25. 25.
    Leonard, N.E., Stability of a Bottom-heavy Underwater Vehicle, Automatica J. IFAC, 1997, vol. 33, pp. 331–346.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lewis, D., Marsden, J., Montgomery, R. and Ratiu, T., The Hamiltonian structure for dynamic free boundary problems, Phys. D, 1986, vol. 18, pp. 391–404.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Marsden, J. and Weinstein, A., Coadjoint Orbits, Vortices, and Clebsch Variables for Incompressible Fluids, Phys. D, 1983, vol. 7, pp. 305–323.CrossRefMathSciNetGoogle Scholar
  28. 28.
    Marsden, J.E., Misiołlek, G., Ortega, J.-P., Perlmutter, M., and Ratiu, T.S., Hamiltonian Reduction by Stages, Lecture Notes in Mathematics, vol. 1913, Berlin: Springer, 2007.zbMATHGoogle Scholar
  29. 29.
    Marsden, J.E. and Perlmutter, M., The Orbit Bundle Picture of Cotangent Bundle Reduction, C. R. Math. Acad. Sci. Soc. R. Can., 2000, vol. 22, pp. 35–54.MathSciNetGoogle Scholar
  30. 30.
    Marsden, J.E. and Ratiu, T.S., Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, vol. 17, New York: Springer, 1994.zbMATHGoogle Scholar
  31. 31.
    Milne-Thomson, L. Theoretical Hydrodynamics. Fifth edition, revised and enlarged. London: MacMillan and Co. Ltd., 1968.zbMATHGoogle Scholar
  32. 32.
    Montgomery, R., The Bundle Picture in Mechanics, PhD thesis, UC Berkeley, 1986;
  33. 33.
    Ovsienko, V.Y. and Khesin, B.A., The Super Korteweg-de Vries Equation as an Euler Equation, Funktsional. Anal. i Prilozhen., 1987, vol. 21, pp. 81–82.zbMATHMathSciNetGoogle Scholar
  34. 34.
    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., 2005, vol. 10, pp. 1–14.zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Sternberg, S., Minimal Coupling and the Symplectic Mechanics of a Classical Particle in the Presence of a Yang-Mills Field, Proc. Nat. Acad. Sci. U.S.A., 1977, vol. 74, pp. 5253–5254.zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Streater, R. F., The Representations of the Oscillator Group, Comm. Math. Phys., 1967, vol. 4, pp. 217–236.zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Vankerschaver, J., Kanso, E. and Marsden, J.E., The Geometry and Dynamics of Interacting Rigid Bodies and Point Vortices, J. Geom. Mech., 2009, vol. 1, pp. 223–266.zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Vizman, C., Geodesics on Extensions of Lie Groups and Stability: the Superconductivity Equation, Phys. Lett. A, 2001, vol. 284, pp. 23–30.zbMATHCrossRefMathSciNetGoogle Scholar

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

Personalised recommendations