Regular and Chaotic Dynamics

, Volume 16, Issue 3–4, pp 223–244 | Cite as

Poisson pencils, algebraic integrability, and separation of variables

Special Issue: Algebraic Integrability

Abstract

In this paper we review a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the bihamiltonian structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.

Keywords

Hamilton-Jacobi equations bihamiltonian manifolds separation of variables generalized Toda lattices 

MSC2010 numbers

14H70 37J35 37K10 70H20 

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References

  1. 1.
    Dubrovin, B.A., Krichever, I. M., and Novikov, S.P., Integrable Systems: 1, in Dynamical Systems IV: Symplectic Geometry and Its Applications, V. I. Arnol’d, S. P. Novikov (Eds.), Encyclopaedia Math. Sci., vol. 4, Berlin: Springer, 2001, pp. 177–332.Google Scholar
  2. 2.
    Sklyanin, E.K., Separations of Variables: New Trends, Progr. Theoret. Phys. Suppl., 1995, vol. 118, pp. 35–60.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Benenti, S., Intrinsic Characterization of the Variable Separation in the Hamilton-Jacobi Equation, J. Math. Phys. 1997, vol. 38, no. 12, pp. 6578–6602.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Adams, M. R., Harnad, J., and Hurtubise, J., Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras, Comm. Math. Phys., 1993, vol. 155, pp. 385–413.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hurtubise, J., Separation of Variables and the Geometry of Jacobians, SIGMA Symmetry Integrability Geom. Methods Appl., 2007, vol. 3, Paper 017, 14 pp. (electronic).Google Scholar
  6. 6.
    Tsiganov, A.V., On Bi-Integrable Natural Hamiltonian Systems on the Riemannian Manifolds, arXiv:1006.3914v2, 2010.Google Scholar
  7. 7.
    Waksjö, C. and Rauch-Wojciechowski, S., How To Find Separation Coordinates for the Hamilton-Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems, Math. Phys. Anal. Geom., 2003, vol. 6, no. 4, pp. 301–348.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Babelon, O. and Viallet, C.-M., Hamiltonian Structures and Lax Equations, Phys. Lett. B, 1990, vol. 237, pp. 411–416.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Marshall, I. D., The Kowalevski Top: Its r-Matrix Interpretation and Bi-Hamiltonian Formulation, Comm. Math. Phys., 1998, vol. 191, no. 3, pp. 723–734.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Reyman, A.G. and Semenov-Tian-Shansky, M.A., Compatible Poisson Structures for Lax Equations: An r-Matrix Approach, Phys. Lett. A, 1988, vol. 130, nos. 8–9, pp. 456–460.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Tsiganov, A.V., A Family of the Poisson Brackets Compatible with the Sklyanin Bracket, J. Phys. A, 2007, vol. 40, no. 18, pp. 4803–4816.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Brouzet, R., Caboz, R., Rabenivo, J., and Ravoson, V., Two Degrees of Freedom Quasi-Bi-Hamiltonian Systems, J. Phys. A, 1996, vol. 29, pp. 2069–2076.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Morosi, C. and Tondo, G., Quasi-Bi-Hamiltonian Systems and Separability, J. Phys. A, 1997, vol. 30, no. 8, pp. 2799–2806.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Błaszak, M., Bi-Hamiltonian Separable Chains on Riemannian Manifolds, Phys. Lett. A, 1998, vol. 243, pp. 25–32.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Falqui, G., Magri, F., and Tondo, G., Reduction of Bi-Hamiltonian Systems and separation of Variables: An Example from the Boussinesq Hierarchy, Teoret. Mat. Fiz., 2000, vol. 122, no. 2, pp. 212–230 [Theoret. and Math. Phys., 2000, vol. 122, no. 2, pp. 176–192].MATHMathSciNetGoogle Scholar
  16. 16.
    Błaszak, M., Theory of Separability of Multi-Hamiltonian Chains, J. Math. Phys., 1999, vol. 40, pp. 5725–5738.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ibort, A., Magri, F., and Marmo, G., Bihamiltonian Structures and Stäckel Separability, J. Geom. Phys., 2000, vol. 33, nos. 3–4, pp. 210–228.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Falqui, G., Magri, F., Pedroni, M., and Zubelli, J.-P., A Bi-Hamiltonian Theory for Stationary KdV Flows and Their Separability, Regul. Chaotic Dyn., 2000, vol. 5, no. 1, pp. 33–52.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Morosi, C. and Tondo, G., The Quasi-Bi-Hamiltonian Formulation of the Lagrange Top, J. Phys. A, 2002, vol. 35, no. 7, pp. 1741–1750.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Pedroni, M., Bi-Hamiltonian Aspects of the Separability of the Neumann System, Teoret. Mat. Fiz., 2002, vol. 133, no. 3, pp. 475–484 [Theoret. and Math. Phys., 2002, vol. 133, no. 3, pp. 1722–1729].MathSciNetGoogle Scholar
  21. 21.
    Falqui, G. and Pedroni, M., Separation of Variables for Bi-Hamiltonian Systems, Math. Phys. Anal. Geom., 2003, vol. 6, no. 2, pp. 139–179.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Bartocci, C., Falqui, G., and Pedroni, M., A Geometric Approach to the Separability of the Neumann-Rosochatius System, Differential Geom. Appl., 2004, vol. 21, no. 3, pp. 349–360.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Falqui, G. and Musso, F., Gaudin Models and Bending Flows: A Geometrical Point of View, J. Phys. A, 2003, vol. 36, pp. 11655–11676.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Falqui, G. and Musso, F., On Separation of Variables for Homogeneous SL(r) Gaudin Systems, Math. Phys. Anal. Geom., 2006, vol. 9, no. 3, pp. 233–262.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Adler, M., van Moerbeke, P., and Vanhaecke, P., Algebraic Integrability, Painlevé Geometry and Lie Algebras, Ergeb. Math. Grenzgeb. (3), vol. 47, Berlin-Heidelberg: Springer, 2004.Google Scholar
  26. 26.
    Veselov, A.P. and Novikov, S.P., Poisson Brackets and Complex Tori, in Algebraic Geometry and Its Applications, Tr. Mat. Inst. Steklova, 1984, vol. 165, pp. 49–61 [Proc. Steklov Inst. Math., 1985, vol. 165, pp. 53–65].MATHMathSciNetGoogle Scholar
  27. 27.
    Krichever, I. M. and Phong, D.H., On the Integrable Geometry of Soliton Equations and N = 2 Supersymmetric Gauge theories, J. Differential Geom., 1997, vol. 45, no. 2, pp. 349–389.MATHMathSciNetGoogle Scholar
  28. 28.
    Krichever, I. M. and Phong, D.H., Symplectic Forms in the Theory of Solitons, in Surveys in Differential Geometry: Integral Systems, Surv. Differ. Geom., vol. 4, Boston, MA: Int. Press, 1998, pp. 239–313.Google Scholar
  29. 29.
    Falqui, G. and Pedroni, M., Gel’fand-Zakharevich Systems and Algebraic Integrability: The Volterra Lattice Revisited, Regul. Chaotic Dyn., 2005, vol. 10, no. 4, pp. 399–412.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Deift, P. A., Li, L.-C., Nanda, T., and Tomei, C., The Toda Lattice on a Generic Orbit is Integrable, Comm. Pure Appl. Math., 1984, vol. 39, pp. 183–232.CrossRefMathSciNetGoogle Scholar
  31. 31.
    Vaisman, I., Lectures on the Geometry of Poisson Manifolds, Progr. Math., vol. 118, Boston: Birkhäuser, 1994.MATHGoogle Scholar
  32. 32.
    Weinstein, A., The Local Structure of Poisson Manifolds, J. Differential Geom., 1983, vol. 18, no. 3, pp. 523–557; (see also: Errata and addenda, J. Differential Geom., 1985, vol. 22, no. 2, p. 255).MATHMathSciNetGoogle Scholar
  33. 33.
    Gel’fand, I.M. and Zakharevich, I., On the Local Geometry of a Bi-Hamiltonian Structure, in The Gel’fand Mathematical Seminars, 1990–1992, L. Corwin, I. M. Gelfand, J. Lepowsky (Eds.), Boston: Birkhäuser, 1993, pp. 51–112.Google Scholar
  34. 34.
    Gel’fand, I.M. and Zakharevich, I., Webs, Lenard Schemes, and the Local Geometry of Bi-Hamiltonian Toda and Lax Structures, Selecta Math. (N. S.), 2000, vol. 6, no. 2, pp. 131–183.CrossRefMathSciNetGoogle Scholar
  35. 35.
    Magri, F. and Morosi, C., A Geometrical Characterization of Integrable Hamiltonian Systems through the Theory of Poisson-Nijenhuis Manifolds, Quaderno, S/19, University of Milan, 1984.Google Scholar
  36. 36.
    Magri, F., Geometry and Soliton Equations, in La Mécanique Analytique de Lagrange et son héritage, Atti Acc. Sci. Torino Suppl., 1990, vol. 124, pp. 181–209.Google Scholar
  37. 37.
    Degiovanni, L. and Magnano, G., Tri-Hamiltonian Vector Fields, Spectral Curves, and Separation Coordinates, Rev. Math. Phys., 2002, vol. 14, pp. 1115–1163.MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Harnad, J. and Hurtubise, J., Multi-Hamiltonian Structures for r-Matrix Systems, J. Math. Phys., 2008, vol. 49, no. 6, 062903, 21 pp.Google Scholar
  39. 39.
    Magri, F. and Marsico, T., Some Developments of the Concept of Poisson Manifolds in the Sense of A. Lichnerowicz, in Gravitation, Electromagnetism, and Geometrical Structures, G. Ferrarese (Ed.), Bologna: Pitagora, 1996, pp. 207–222.Google Scholar
  40. 40.
    Marsico, T., Una caratterizzazione geometrica dei sistemi che ammettono rappresentazione alla Lax estesa, Ph. D. thesis, Università di Milano, 1996.Google Scholar
  41. 41.
    Kupershmidt, B. A., Discrete Lax Equations and Differential-Difference Calculus, Asterisque, 1985, vol. 123, pp. 212–245.MathSciNetGoogle Scholar
  42. 42.
    Carlet, G., The Extended Bigraded Toda Hierarchy, J. Phys. A, 2006, vol. 39, pp. 9411–9435.MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Meucci, A., Toda Equations, Bi-Hamiltonian Systems, and Compatible Lie Algebroids, Math. Phys. Anal. Geom., 2001, vol. 4, no. 2, pp. 131–146.MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Damianou, P.A., Magri, F., A Gentle (without Chopping) Approach to the Full Kostant-Toda Lattice, SIGMA Symmetry Integrability Geom. Methods Appl., 2005, vol. 1, Paper 010, 12 pp. (electronic).Google Scholar
  45. 45.
    Falqui, G. and Pedroni, M., On a Poisson Reduction for Gel’fand-Zakharevich Manifolds, Rep. Math. Phys., 2002, vol. 50, no. 3, pp. 395–407.MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Marciniak, K. and Błaszak, M., Dirac Reduction Revisited, J. Nonlinear Math. Phys., 2003, vol. 10, no. 4, pp. 451–463.MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Sklyanin, E.K., Separations of Variables in the Classical Integrable SL(3) Magnetic Chain, Comm. Math. Phys., 1992, vol. 150, no. 1, pp. 181–192.MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Dubrovin, B.A. and Diener, P., Algebraic Geometrical Darboux Coordinates in R-Matrix Formalism, SISSA preprint 88/94/FM, 1994.Google Scholar
  49. 49.
    Adams, M.R., Harnad, J., and Hurtubise, J., Darboux Coordinates on Coadjoint Orbits of Lie Algebras, Lett. Math. Phys., 1997, vol. 40, pp. 41–57.MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Kuznetsov, V. B., Nijhoff, F.W., and Sklyanin, E.K., Separation of Variables for the Ruijsenaars System, Comm. Math. Phys., 1997, vol. 189, no. 3, pp. 855–877.MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Tsiganov, A.V., On the Invariant Separated Variables, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 307–326.MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Ercolani, N.M., Flaschka, H., and Singer, S., The Geometry of the Full Kostant-Toda Lattice, in Integrable Systems: The Verdier Memorial Conference: Actes du Colloque International de Luminy (1991), O. Babelon et al. (Eds.), Progr. Math., vol. 115, Boston: Birkhäuser, 1993, pp. 181–226.Google Scholar
  53. 53.
    Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.MATHGoogle Scholar
  54. 54.
    Magri, F., Falqui, G., and Pedroni, M., The Method of Poisson Pairs in the Theory of Nonlinear PDEs, in Direct and Inverse Methods in Nonlinear Evolution Equations, Lecture Notes in Phys., vol. 632, Berlin: Springer, 2003, pp. 85–136.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano-BicoccaMilanoItaly
  2. 2.Dipartimento di Ingegneria dell’Informazione e Metodi MatematiciUniversità di BergamoDalmine (BG)Italy

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