Journal of Nonlinear Science

, Volume 24, Issue 1, pp 1–37 | Cite as

A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

  • Joris Vankerschaver
  • Melvin Leok


In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge–Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.


Point vortices Hopf fibration Symplectic integration Variational methods 

Mathematics Subject Classification

37M15 76B47 70H03 



We are very grateful to the referees of this paper, whose comments and observations significantly improved our exposition.

We would like to dedicate this paper to the memory of Hassan Aref, whose kind encouragement and insightful remarks at the 2010 SIAM-SEAS meeting at the University of North Carolina, Charlotte, provided the initial stimulus for this work. Furthermore, we would like to thank J.D. Brown, C. Burnett, B. Cheng, F. Gay-Balmaz, M. Gotay, D. Holm, E. Kanso, S.D. Kelly, P. Newton, T. Ohsawa, B. Shashikanth and A. Stern for stimulating discussions and helpful remarks.

M.L. and J.V. are partially supported by NSF grants DMS-1010687, CMMI-1029445, and DMS-1065972. J.V. is on leave from a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO-Vlaanderen). This work is supported by the irses project geomech (nr. 246981) within the 7th European Community Framework Programme.


  1. Aref, H.: Point vortex dynamics: a classical mathematics playground. J. Math. Phys. 48, 065401 (2007) CrossRefMathSciNetGoogle Scholar
  2. Aref, H.: Relative equilibria of point vortices and the fundamental theorem of algebra. Proc. R. Soc. A, Math. Phys. Eng. Sci. 467, 2168–2184 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  3. Benettin, G., Giorgilli, A.: On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74, 1117–1143 (1994) CrossRefzbMATHMathSciNetGoogle Scholar
  4. Birkhoff, G.D.: Dynamical Systems. American Mathematical Society Colloquium Publications, vol. IX. American Mathematical Society, Providence (1966). With an addendum by Jurgen Moser zbMATHGoogle Scholar
  5. Boatto, S., Koiller, J.: Vortices on closed surfaces. Preprint. arXiv:0802.4313v1 (2008)
  6. Bogomolov, V.A.: Dynamics of vorticity at a sphere. Fluid Dyn. 12, 863–870 (1977). doi: 10.1007/BF01090320 CrossRefGoogle Scholar
  7. Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Absolute and relative choreographies in the problem of point vortices moving on a plane. Regul. Chaotic Dyn. 9, 101–111 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  8. Bou-Rabee, N., Marsden, J.E.: Hamilton–Pontryagin integrators on Lie groups. I. Introduction and structure-preserving properties. Found. Comput. Math. 9, 197–219 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  9. Boyland, P., Stremler, M., Aref, H.: Topological fluid mechanics of point vortex motions. Physica D 175, 69–95 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  10. Brown, J.D.: Midpoint rule as a variational-symplectic integrator: Hamiltonian systems. Phys. Rev. D 73, 024001 (2006) CrossRefGoogle Scholar
  11. Bruveris, M., Ellis, D.C.P., Holm, D.D., Gay-Balmaz, F.: Un-reduction. J. Geom. Mech. 3, 363–387 (2011) zbMATHMathSciNetGoogle Scholar
  12. Cendra, H., Marsden, J.E.: Lin constraints, Clebsch potentials and variational principles. Physica D 27, 63–89 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  13. Chamoun, G., Kanso, E., Newton, P.K.: Von Kármán vortex streets on the sphere. Phys. Fluids 21, 116603 (2009) CrossRefGoogle Scholar
  14. Chapman, D.M.F.: Ideal vortex motion in two dimensions: symmetries and conservation laws. J. Math. Phys. 19, 1988–1992 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  15. Chartier, P., Darrigrand, E., Faou, E.: A regular fast multipole method for geometric numerical integrations of Hamiltonian systems. BIT Numer. Math. 50, 23–40 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  16. Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973) CrossRefMathSciNetGoogle Scholar
  17. Cotter, C.J., Holm, D.D.: Continuous and discrete Clebsch variational principles. Found. Comput. Math. 9, 221–242 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  18. Engø, K., Faltinsen, S.: Numerical integration of Lie–Poisson systems while preserving coadjoint orbits and energy. SIAM J. Numer. Anal. 39, 128–145 (2002) CrossRefGoogle Scholar
  19. Faddeev, L., Jackiw, R.: Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett. 60, 1692–1694 (1988) CrossRefzbMATHMathSciNetGoogle Scholar
  20. Frankel, T.: The Geometry of Physics: An Introduction, 2nd edn. Cambridge University Press, Cambridge (2004) Google Scholar
  21. Gotay, M.: Presymplectic manifolds, geometric constraint theory and the Dirac-Bergmann theory of constraints. PhD thesis, University of Maryland (1979) Google Scholar
  22. Hairer, E.: Backward analysis of numerical integrators and symplectic methods. Ann. Numer. Math. 1, 107–132 (1994) zbMATHMathSciNetGoogle Scholar
  23. Hairer, E., Lubich, C.: The life-span of backward error analysis for numerical integrators. Numer. Math. 76, 441–462 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  24. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, 1st edn. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2002) CrossRefzbMATHGoogle Scholar
  25. Khesin, B.: Symplectic structures and dynamics on vortex membranes. Mosc. Math. J. 12, 413–434 (2012). 461–462 zbMATHMathSciNetGoogle Scholar
  26. Kidambi, R., Newton, P.K.: Motion of three point vortices on a sphere. Physica D 116, 143–175 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  27. Kimura, Y., Okamoto, H.: Vortex motion on a sphere. J. Phys. Soc. Jpn. 56, 4203–4206 (1987) CrossRefMathSciNetGoogle Scholar
  28. Kobilarov, M., Marsden, J.: Discrete geometric optimal control on Lie groups. IEEE Trans. Robot. 27, 641–655 (2011) CrossRefGoogle Scholar
  29. Kostant, B.: Minimal coadjoint orbits and symplectic induction. In: The Breadth of Symplectic and Poisson Geometry. Progr. Math., vol. 232, pp. 391–422. Birkhäuser, Boston (2005) CrossRefGoogle Scholar
  30. Lamb, H.: Hydrodynamics. Dover Publications, New York (1945). Reprint of the 1932 Cambridge University Press edition Google Scholar
  31. Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem. Comput. Methods Appl. Mech. Eng. 196, 2907–2924 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  32. Lee, T., Leok, M., McClamroch, N.H.: Lagrangian mechanics and variational integrators on two-spheres. Int. J. Numer. Methods Eng. 79, 1147–1174 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  33. Lemaître, G.: Quaternions et espace elliptique. Pont. Acad. Sci. Acta 12, 57–78 (1948) MathSciNetGoogle Scholar
  34. Leok, M., Zhang, J.: Discrete Hamiltonian variational integrators. IMA J. Numer. Anal. 31, 1497–1532 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  35. Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. Z. Angew. Math. Mech. 88, 677–708 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  36. Lim, C., Montaldi, J., Roberts, M.: Relative equilibria of point vortices on the sphere. Physica D 148, 97–135 (2001) CrossRefMathSciNetGoogle Scholar
  37. Ma, Z., Rowley, C.W.: Lie-Poisson integrators: a Hamiltonian, variational approach. Int. J. Numer. Methods Eng. 82, 1609–1644 (2010) zbMATHMathSciNetGoogle Scholar
  38. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002) zbMATHGoogle Scholar
  39. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  40. McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1998) zbMATHGoogle Scholar
  41. Milne-Thomson, L.: Theoretical Hydrodynamics, 5th edn. MacMillan and Co. Ltd., London (1968). Revised and enlarged edition zbMATHGoogle Scholar
  42. Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs, vol. 91. American Mathematical Society, Providence (2002) zbMATHGoogle Scholar
  43. Moser, J., Veselov, A.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217–243 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  44. Newton, P.K.: The N-Vortex Problem. Analytical Techniques. Applied Mathematical Sciences, vol. 145. Springer, New York (2001) CrossRefGoogle Scholar
  45. Newton, P.K., Sakajo, T.: Point vortex equilibria and optimal packings of circles on a sphere. Proc. R. Soc. A, Math. Phys. Eng. Sci. 467, 1468–1490 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  46. Novikov, S.P.: The Hamiltonian formalism and a many-valued analogue of Morse theory. Russ. Math. Surv. 37, 1 (1982) CrossRefzbMATHGoogle Scholar
  47. Oh, Y.-G.: Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle. J. Differ. Geom. 46, 499–577 (1997) zbMATHGoogle Scholar
  48. Oliphant, T.E.: Python for scientific computing. Comput. Sci. Eng. 9, 10–20 (2007) CrossRefGoogle Scholar
  49. Onsager, L.: Statistical hydrodynamics. Nuovo Cimento 6, 279–287 (1949) CrossRefMathSciNetGoogle Scholar
  50. Owren, B., Welfert, B.: The Newton iteration on Lie groups. BIT Numer. Math. 40, 121–145 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  51. Pekarsky, S., Marsden, J.E.: Point vortices on a sphere: stability of relative equilibria. J. Math. Phys. 39, 5894–5907 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  52. Polvani, L.M., Dritschel, D.G.: Wave and vortex dynamics on the surface of a sphere. J. Fluid Mech. 255, 35–64 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  53. Pullin, D.I., Saffman, P.G.: Long-time symplectic integration: the example of four-vortex motion. Proc. Math. Phys. Sci. 432, 481–494 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  54. Reich, S.: Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36, 1549–1570 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  55. Rowley, C., Marsden, J.: Variational integrators for degenerate Lagrangians, with application to point vortices. In: Proceedings of the 41st IEEE Conference on Decision and Control, vol. 2, pp. 1521–1527 (2002) Google Scholar
  56. Saffman, P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York (1992) zbMATHGoogle Scholar
  57. Sakajo, T.: Motion of a vortex sheet on a sphere with pole vortices. Phys. Fluids 16, 717–727 (2004) CrossRefMathSciNetGoogle Scholar
  58. Sakajo, T.: Non-self-similar, partial, and robust collapse of four point vortices on a sphere. Phys. Rev. E 78, 016312 (2008) CrossRefMathSciNetGoogle Scholar
  59. Shashikanth, B.N.: Vortex dynamics in \(\Bbb{R}^{4}\). J. Math. Phys. 53, 013103, 21 (2012) CrossRefMathSciNetGoogle Scholar
  60. Soulière, A., Tokieda, T.: Periodic motions of vortices on surfaces with symmetry. J. Fluid Mech. 460, 83–92 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  61. Urbantke, H.K.: The Hopf fibration—seven times in physics. J. Geom. Phys. 46, 125–150 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  62. von Helmholtz, H.: Über Integrale der hydro-dynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858) CrossRefzbMATHGoogle Scholar
  63. Woodhouse, N.M.J.: Geometric Quantization, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1992) zbMATHGoogle Scholar
  64. Zhong, G., Marsden, J.E.: Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133, 134–139 (1988) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA
  2. 2.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations