Regular and Chaotic Dynamics

, Volume 15, Issue 4–5, pp 532–550 | Cite as

Poisson structures for geometric curve flows in semi-simple homogeneous spaces

Special Issue: Valery Vasilievich Kozlov-60


We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.

Key words

moving frame Poisson structure homogeneous space invariant curve flow differential invariant invariant variational bicomplex 

MSC2000 numbers

22F05 35A30 35Q53 53A55 58A20 53D17 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chou, K.-S., and Qu, C., Integrable Equations Arising from Motions of Plane Curves, Physica D, 2002, vol. 162, pp. 9–33.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chou, K.-S., and Qu, C.-Z., Integrable Equations Arising from Motions of Plane Curves II, J. Nonlinear Sci., 2003, vol. 13, pp. 487–517.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Doliwa, A., and Santini, P.M., An Elementary Geometric Characterization of the Integrable Motions of a Curve, Phys. Lett. A, 1994, vol. 185, pp. 373–384.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dubrovin, B.A., and Novikov, S.P., On Poisson Brackets of Hydrodynamic Type, Sov. Math. Dokl., 1984, vol. 30, pp. 651–654.MATHGoogle Scholar
  5. 5.
    Dubrovin, B.A., and Novikov, S.P., Hydrodynamics of Weakly Deformed Soliton Lattices. Differential Geometry and Hamiltonian Theory, Russian Math. Surveys, 1989, vol. 44:6, pp. 35–124.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fels, M., and Olver, P.J., Moving Coframes. II. Regularization and Theoretical Foundations, Acta Appl. Math., 1999, vol. 55, pp. 127–208.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Green, M.L., The Moving Frame, Differential Invariants and Rigidity Theorems for Curves in Homogeneous Spaces, Duke Math. J., 1978, vol. 45, pp. 735–779.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Guggenheimer, H.W., Differential Geometry, New York: McGraw-Hill, 1963.MATHGoogle Scholar
  9. 9.
    Hasimoto, H., A Soliton on a Vortex Filament, J. Fluid Mech., 1972, vol. 51, pp. 477–485.MATHCrossRefGoogle Scholar
  10. 10.
    Hubert, E., Differential Invariants of a Lie Group Action: Syzygies on a Generating Set, J. Symb. Comp., 2009, vol. 44, pp. 382–416.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ivey, T., Integrable Geometric Evolution Equations for Curves, Contemp. Math., 2001, vol. 285, pp. 71–84.MathSciNetGoogle Scholar
  12. 12.
    Kirillov, A.A., Merits and Demerits of the Orbit Method, Bull. Amer. Math. Soc., 1999, vol. 36, pp. 433–488.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kogan, I.A., and Olver, P.J., Invariant Euler-Lagrange Equations and the Invariant Variational Bicomplex, Acta Appl. Math., 2003, vol. 76, pp. 137–193.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mansfield, E.L., and van der Kamp, P.E., Evolution of Curvature Invariants and Lifting Integrability, J. Geom. Phys., 2006, vol. 56, pp. 1294–1325.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Marí Beffa, G., Poisson Brackets Associated to the Conformal Geometry of Curves, Trans. Amer. Math. Soc., 2005, vol. 357, pp. 2799–2827.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Marí Beffa, G., Poisson Geometry of Differential Invariants of Curves in Some Nonsemisimple Homogeneous Spaces, Proc. Amer. Math. Soc., 2006, vol. 134, pp. 779–791.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Marí Beffa, G., Hamiltonian Structures on the Space of Differential Invariants of Curves in Flat Semisimple Homogenous Manifolds, Asian J. Math., 2008, vol. 12, pp. 1–33.MATHMathSciNetGoogle Scholar
  18. 18.
    Marí Beffa, G., On Bi-Hamiltonian Flows and Their Realizations as Curves in Real Semisimple Homogeneous Manifolds, Pacific J. Math., to appear.Google Scholar
  19. 19.
    Marí Beffa, G., Sanders, J.A., and Wang, J.P., Integrable systems in three-dimensional Riemannian geometry, J. Nonlinear Sci., 2002, vol. 12, pp. 143–167.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Marsden, J.E., and Ratiu, T., Reduction of Poisson manifolds, Lett. Math. Phys., 1986, vol. 11, pp. 161–169.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Nakayama, K., Segur, H., and Wadati, M., Integrability and Motion of Curves, Phys. Rev. Lett., 1992, vol. 69, pp. 2603–2606.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Olver, P.J., Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, New York: Springer-Verlag, 1993.MATHGoogle Scholar
  23. 23.
    Olver, P.J., Generating Differential Invariants, J. Math. Anal. Appl., 2007, vol. 333, pp. 450–471.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Olver, P.J., Invariant Submanifold Flows, J. Phys. A, 2008, vol. 41, 344017.CrossRefMathSciNetGoogle Scholar
  25. 25.
    Olver, P.J., Differential Invariant Algebras, Comtemp. Math., to appear.Google Scholar
  26. 26.
    Weiss, J., The Painlevé Property for Partial Differential Equations. II. Bäcklund Transformation, Lax Pairs, and the Schwarzian Derivative, J. Math. Phys., 1983, vol. 24, pp. 1405–1413.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations