Separation of Variables for Bi-Hamiltonian Systems

  • Gregorio Falqui
  • Marco Pedroni


We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called ωN manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the ωN manifold itself. We apply these results to bi-Hamiltonian systems of the Gel'fand–Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.

Hamilton–Jacobi equations separation of variables Nijenhuis structures bi-Hamiltonian manifolds 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, M. R., Harnad, J. and Hurtubise, J.: Darboux coordinates and Liouville-Arnold integration in loop algebras, Comm. Math. Phys. 155 (1993), 385–413.Google Scholar
  2. 2.
    Adams, M. R., Harnad, J. and Hurtubise, J.: Darboux coordinates on coadjoint orbits of Lie algebras, Lett. Math. Phys. 40 (1997), 41–57.Google Scholar
  3. 3.
    Adler, M. and van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves, Adv. Math. 38 (1980), 267–317.Google Scholar
  4. 4.
    Benenti, S.: Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 (1997), 6578–6602.Google Scholar
  5. 5.
    Błaszak, M.: On separability of bi-Hamiltonian chain with degenerated Poisson structures, J. Math. Phys. 39 (1998), 3213–3235.Google Scholar
  6. 6.
    Bruce, A. T., McLenaghan, R. G. and Smirnov, R. G.: A systematic study of the Toda lattice in the context of the Hamilton-Jacobi theory, Z. Angew. Math. Phys. 52 (2001), 171–190.Google Scholar
  7. 7.
    Crampin, M., Sarlet,W. and Thompson, G.: Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors, J. Phys. A 33 (2000), 8755–8770.Google Scholar
  8. 8.
    Degiovanni, L. and Magnano, G.: Tri-hamiltonian vector fields, spectral curves and separation coordinates, Rev. Math. Phys. 14 (2002), 1115–1163.Google Scholar
  9. 9.
    Dubrovin, B. A., Krichever, I. M. and Novikov, S. P.: Integrable systems. I, In: V. I. Arnol'd and S. P. Novikov (eds), Encyclopaedia of Math. Sci. 4 (Dynamical Systems IV), Springer-Verlag, Berlin, 1990.Google Scholar
  10. 10.
    Falqui, G., Magri, F. and Pedroni, M.: in preparation.Google Scholar
  11. 11.
    Falqui, G., Magri, F., Pedroni, M. and Zubelli, J. P.: A bi-Hamiltonian theory for stationary KdV flows and their separability, Regul. Chaotic Dynamics 5 (2000), 33–52.Google Scholar
  12. 12.
    Falqui, G., Magri, F. and Pedroni, M.: Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys. 8 (2001), suppl. (Proc. NEEDS99), 118–127.Google Scholar
  13. 13.
    Falqui, G., Magri, F. and Tondo, G.: Bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy, Theor. Math. Phys. 122 (2000), 176–192.Google Scholar
  14. 14.
    Falqui, G. and Pedroni, M.: On a Poisson reduction for Gel'fand-Zakharevich manifolds, Rep. Math. Phys. 50 (2002), 395–407.Google Scholar
  15. 15.
    Flaschka, H.: Integrable systems and torus actions, In: O. Babelon et al. (eds), Lectures on Integrable Systems, World Scientific, Singapore, 1994, pp. 43–101.Google Scholar
  16. 16.
    Flaschka, H. and McLaughlin, D. W.: Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theor. Phys. 55 (1976), 438–456.Google Scholar
  17. 17.
    Frölicher, A. and Nijenhuis, A.: Theory of vector-valued differential forms, Proc. Ned. Acad. Wetensch. Ser. A 59 (1956), 338–359.Google Scholar
  18. 18.
    Gel'fand, I. M. and Zakharevich, I.: On the local geometry of a bi-Hamiltonian structure, In: L. Corwin et al. (eds), The Gel'fand Mathematical Seminars 1990-1992, Birkhäuser, Boston, 1993, pp. 51–112.Google Scholar
  19. 19.
    Gel'fand, I. M. and Zakharevich, I.: Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures, Selecta Math. (NS) 6 (2000), 131–183.Google Scholar
  20. 20.
    Gorsky, A., Nekrasov, N. and Rubtsov, V.: Hilbert schemes, separated variables, and D-branes, Comm. Math. Phys. 222 (2001), 299–318.Google Scholar
  21. 21.
    Harnad, J. and Hurtubise, J. C.: Multi-Hamiltonian structures for r-matrix systems, mathph/ 0211076.Google Scholar
  22. 22.
    Hitchin, N. J.: The moduli space of complex Lagrangian submanifolds, Asian J.Math. 3 (1999), 77–91.Google Scholar
  23. 23.
    Hurtubise, J. C. and Kjiri, M.: Separating coordinates for the generalized Hitchin systems and the classical r-matrices, Comm. Math. Phys. 210 (2000), 521–540.Google Scholar
  24. 24.
    Ibort, A., Magri, F. and Marmo, G.: Bihamiltonian structures and Stäckel separability, J. Geom. Phys. 33 (2000), 210–228.Google Scholar
  25. 25.
    Kalnins, E. G.: Separation of Variables for Riemannian Spaces of Constant Curvature, Longman, Harlow; Wiley, New York, 1986.Google Scholar
  26. 26.
    Kosmann-Schwarzbach, Y. and Magri, F.: Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Theor. 53 (1990), 35–81.Google Scholar
  27. 27.
    Magri, F.: Geometry and soliton equations. In: La mécanique analytique de Lagrange et son héritage, Atti Accad. Sci. Torino Suppl. 124 (1990), 181–209.Google Scholar
  28. 28.
    Magri, F.: Eight lectures on Integrable Systems, In: Y. Kosmann-Schwarzbach et al. (eds), Integrability of Nonlinear Systems, Lecture Notes in Phys. 495, Springer-Verlag, Berlin, 1997, pp. 256–296.Google Scholar
  29. 29.
    Magri, F. and Morosi, C.: Sulla relazione tra varietà bihamiltoniane ed i problemi spettrali della teoria dello scattering inverso, Atti 8o Congresso AIMETA (Torino, 1986), Vol. II, pp. 675–679 (in Italian).Google Scholar
  30. 30.
    Magri, F., Morosi, C. and Ragnisco, O.: Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications, Comm. Math. Phys. 99 (1985), 115–140.Google Scholar
  31. 31.
    Marsden, J. E. and Ratiu, T.: Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), 161–169.Google Scholar
  32. 32.
    Morosi, C. and Pizzocchero, L.: R-matrix theory, formal Casimirs and the periodic Toda lattice, J. Math. Phys. 37 (1996), 4484–4513.Google Scholar
  33. 33.
    Morosi, C. and Tondo, G.: Quasi-Bi-Hamiltonian systems and separability, J. Phys. A.: Math. Gen. 30 (1997), 2799–2806.Google Scholar
  34. 34.
    Panasyuk, A.: Veronese webs for bihamiltonian structures of higher corank, In: P. Urbànski and J. Grabowski (eds), Poisson Geometry (Warsaw, 1998), Banach Center Publ. 51, Polish Acad. Sci., Warsaw, 2000.Google Scholar
  35. 35.
    Pars, L. A.: A Treatise on Analytical Dynamics, Heinemann, London, 1965.Google Scholar
  36. 36.
    Pedroni, M.: Bi-Hamiltonian aspects of the separability of the Neumann system, Theor. Math. Phys. 133 (2002), 1722–1729.Google Scholar
  37. 37.
    Reyman, A. G. and Semenov-Tian-Shansky, M. A.: Compatible Poisson structures for Lax equations: an r-matrix approach, Phys. Lett. A 130 (1988), 456–460.Google Scholar
  38. 38.
    Reyman, A. G. and Semenov-Tian-Shansky, M. A.: Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable systems, In: V. I. Arnol'd and S. P. Novikov (eds), Encyclopaedya of Math. Sci. 16 (Dynamical Systems VII), Springer-Verlag, Berlin, 1994.Google Scholar
  39. 39.
    Sklyanin, E. K.: Separations of variables: new trends, Progr. Theor. Phys. Suppl. 118 (1995), 35–60.Google Scholar
  40. 40.
    Tsiganov, A. V.: On the invariant separated variables, Regul. Chaotic Dynamics 6 (2001), 307–326.Google Scholar
  41. 41.
    Turiel, F.-J.: Classification locale d'un couple de formes symplectiques Poisson-compatibles, C.R. Acad. Sci. Paris Sér. I Math. 308 (1989), 575–578.Google Scholar
  42. 42.
    Vaisman, I.: Lectures on the Geometry of Poisson Manifolds, Progr.Math., Birkhäuser, Boston, 1994.Google Scholar
  43. 43.
    Veselov, A. P. and Novikov, S. P.: Poisson brackets and complex tori, Proc. Steklov Inst. Math. 3 (1985), 53–65.Google Scholar
  44. 44.
    Woodhouse, N. M. J.: Killing tensors and the separation of the Hamilton-Jacobi equation, Comm. Math. Phys. 44 (1975), 9–38.Google Scholar
  45. 45.
    Yano, K. and Ishihara, S.: Tangent and Cotangent Bundles: Differential Geometry, Marcel Dekker, New York, 1973.Google Scholar
  46. 46.
    Yunbo Zeng and Wen-Xiu Ma, Families of quasi-bi-Hamiltonian systems and separability, J. Math. Phys. 40 (1999), 4452–4473.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Gregorio Falqui
    • 1
  • Marco Pedroni
    • 2
  1. 1.SISSATriesteItaly. e-mail
  2. 2.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

Personalised recommendations