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Classical Separability Theory

  • Maciej Błaszak
Chapter

Abstract

As was analysed in the previous chapter, once we find separation coordinates for a Liouville integrable system, we can integrate the system by quadratures through an appropriate separation relations. The fundamental problem in the Hamilton–Jacobi method is the systematic construction of transformation from some “natural” coordinates to separation coordinates. As was demonstrated in the previous chapter, such coordinates like Cartesian, spherical or cylindrical are separation coordinates only in very special cases. In general, separation coordinates are much less obvious and completely unpredictable. So the question about the existence of a systematic method for the construction of separation coordinates is very important. Indeed, for many decades of development of the separability theory, the method did not exist. Only recently, at the end of the twentieth century, after more than 100 years of efforts, a few different constructive methods were suggested. Obviously, the knowledge of all constants of motion for a given Liouville integrable system is not enough. Some extra information is required.

References

  1. 3.
    Antonowicz, M., Rauch-Wojciechowski, S.: How to construct finite dimensional bi-Hamiltonian systems from soliton equations: Jacobi integrable potentials. J. Math. Phys. 33, 2115 (1992)MathSciNetCrossRefADSGoogle Scholar
  2. 4.
    Antonowicz, M., Fordy, A.P., Rauch-Wojciechowski, S.: Integrable stationary flows: Miura maps and bi-Hamiltonian structures. Phys. Lett. A 124, 143 (1987)MathSciNetCrossRefADSGoogle Scholar
  3. 5.
    Ay, A., Gürses, M., Zheltukhin, K.: Hamiltonian equations in \({\mathbb {R}}^{3}\). J. Math. Phys. 44(12), 5688 (2003)Google Scholar
  4. 6.
    Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems. Cambridge University, Cambridge (2003)CrossRefGoogle Scholar
  5. 23.
    Błaszak, M.: On separability of bi-Hamiltonian chain with degenerate Poisson structures. J. Math. Phys. 39, 3213 (1998)MathSciNetCrossRefADSGoogle Scholar
  6. 24.
    Błaszak, M.: Multi-Hamiltonian theory of dynamical systems. In: Texts and Monographs in Physics. Springer, Berlin (1998)Google Scholar
  7. 25.
    Błaszak, M.: Bi-Hamiltonian separable chains on Riemannian manifolds. Phys. Lett. A 243, 25 (1998)MathSciNetCrossRefADSGoogle Scholar
  8. 26.
    Błaszak, M.: Theory of separability of multi-Hamiltonian chains. J. Math. Phys. 40, 5725 (1999)MathSciNetCrossRefADSGoogle Scholar
  9. 28.
    Błaszak, M.: From bi-Hamiltonian geometry to separation of variables: stationary Harry-Dym and the KdV dressing chain. J. Nonl. Math. Phys. 9(1), 1 (2002)MathSciNetCrossRefGoogle Scholar
  10. 29.
    Błaszak, M.: Presymplectic representation of bi-Hamiltonian chains. J. Phys. A Math. Gen. 37(50), 11971 (2004)MathSciNetCrossRefADSGoogle Scholar
  11. 31.
    Błaszak, M.: Bi-Hamiltonian representation of Stäckel systems. Phys. Rev. E 79, 056607 (2009)MathSciNetCrossRefADSGoogle Scholar
  12. 32.
    Błaszak, M.: Bi-presymplectic representation of Liouville integrable systems and related separability theory. Stud. Appl. Math. 126, 319 (2011)MathSciNetCrossRefGoogle Scholar
  13. 44.
    Błaszak, M., Sergyeyev, A.: Natural coordinates for a class of Benenti systems. Phys. Lett. A 365, 28 (2007)MathSciNetCrossRefADSGoogle Scholar
  14. 45.
    Błaszak, M., Sergyeyev, A.: Generalized Stäckel systems. Phys. Lett. A 375(27), 2617 (2011)MathSciNetCrossRefADSGoogle Scholar
  15. 46.
    Błaszak, M., Goürses, M., Zheltukhin, C.: Bi-presymplectic chains of co-rank 1 and related Liouville integrable systems. J. Phys. A Math. Theor. 42, 285204 (2009)MathSciNetCrossRefADSGoogle Scholar
  16. 60.
    Brouzet, R., Caboz, R., Rabenivo, J.: Quasi-bi-Hamiltonian systems and separability. J. Phys. A Math. Gen. 29, 2069 (1996)CrossRefADSGoogle Scholar
  17. 68.
    Coodonovsky, D.V., Choodonovsky, G.V.: Completely integrable class of mechanical systems connected with Korteweg-de Vries and multicomponent Schrödinger equations. Lett. Nuovo Cimento 22, 47 (1978)MathSciNetCrossRefGoogle Scholar
  18. 70.
    Crampin, M.: Projectively equivalent Riemannian spaces as quasi-bi-Hamiltonian systems. Acta Appl. Math. 77, 237 (2003)MathSciNetCrossRefGoogle Scholar
  19. 72.
    Crampin, M., Sarlet, W.: A class of nonconservative Lagrangian systems on Riemannian manifolds. J. Math. Phys. 42, 4313 (2001)MathSciNetCrossRefADSGoogle Scholar
  20. 73.
    Crampin, M., Sarlet, W.: Bi-quasi-Hamiltonian systems. J. Math. Phys. 43, 2505 (2001)MathSciNetCrossRefADSGoogle Scholar
  21. 77.
    Damianou, P.A.: Multiple Hamiltonian structures for Toda-type systems. J. Math. Phys. 35, 5511 (1994)MathSciNetCrossRefADSGoogle Scholar
  22. 107.
    Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B., Legkin, D.V.: Linear r-matrix algebra for systems separable in parabolic coordinates. Phys. Lett. A 180, 208 (1993)MathSciNetCrossRefADSGoogle Scholar
  23. 108.
    Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B., Tsiganov, A.V.: Linear r-matrix algebra for classical separable systems. J. Phys. A Gen. Math. 27, 567 (1994)MathSciNetCrossRefADSGoogle Scholar
  24. 112.
    Falqui, G., Pedroni, M.: On a Poisson reduction for Gel’fand-Zakharevich manifolds. Rep. Math. Phys. 50, 395 (2002)MathSciNetCrossRefADSGoogle Scholar
  25. 113.
    Falqui, G., Pedroni, M.: Separation of variables for Bi-Hamiltonian systems. Math. Phys. Anal. Geom. 6, 139 (2003)MathSciNetCrossRefGoogle Scholar
  26. 115.
    Falqui, G., Magri, F., Pedroni, M.: Bihamiltonian geometry and separation of variables for Toda lattices. J. Nonlinear Math. Phys. 8(Suppl.), 118 (2001)MathSciNetCrossRefADSGoogle Scholar
  27. 117.
    Fedorov, Yu. N.: Integrable systems, Poisson pencils and hipperelliptic Lax pairs. Regul. Chaotic Dyn. 5, 171 (2000)MathSciNetCrossRefGoogle Scholar
  28. 121.
    Flaschka, H.: The Toda lattice I: Existence of integrals. Phys. Rev. B 9, 1924 (1974)MathSciNetCrossRefADSGoogle Scholar
  29. 123.
    Gel’fand, I.M., Zakharevich, I.: On the local geometry of a bi-Hamiltonian structure. In: The Gel’fand Mathematical Seminars 1990–1992 (eds.) Corwin, L. et. al., p. 51 Birkhäuser, Boston (1993)Google Scholar
  30. 124.
    Gel’fand, I.M., Zakharevich, I.: Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures. Selecta Math. (N.S) 6, 131 (2000)Google Scholar
  31. 141.
    Gümral, X, Nutku, Y.: Poisson structure of dynamical systems with three degrees of freedom. J. Math. Phys. 34, 5691 (1993)MathSciNetCrossRefADSGoogle Scholar
  32. 156.
    Ibort, A., Magri, F., Marmo, G.: Bihamiltonian structures and Stäckel separability. J. Geom. Phys. 33, 210 (2000)MathSciNetCrossRefADSGoogle Scholar
  33. 157.
    Jacobi, C.G.: Vörlesungen über Dynamik, vol. 9, pp. 26–29. Georg Reimer, Berlin (1866)Google Scholar
  34. 173.
    Kosmann-Schwarzbach, Y., Magri, F.: Poisson-Nijenhuis structures. Ann. Inst. H. Poincaré Phys. Theor. 53, 35 (1990)MathSciNetzbMATHGoogle Scholar
  35. 175.
    Kuznetsov, V.B., Nijhoff, F.W., Sklyanin, E.: Separation of variables for the Ruijsenaars system. Commun. Math. Phys. 189, 855 (1997)MathSciNetCrossRefADSGoogle Scholar
  36. 184.
    Lundmark, H.: Higher-dimensional integrable Newton systems with quadratic integrals of motion. Stud. Appl. Math. 110, 257 (2003)MathSciNetCrossRefGoogle Scholar
  37. 186.
    Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156 (1978)MathSciNetCrossRefADSGoogle Scholar
  38. 187.
    Magri, F.: Eight lectures on Integrable Systems. In: Integrability of Nonlinear Systems, Kosmann-Schwarzbach, Y. et al. (eds.), Lecture notes in Physics, vol. 495. Springer, Berlin (1997)Google Scholar
  39. 188.
    Manakov, S.V.: Remarks on the integrals of the Euler equations of the n-dimensional heavy top. Funct. Anal. Appl. 10, 93 (1976)Google Scholar
  40. 189.
    Marciniak, K., Błaszak, M.: Separation of variables in quasi-potential systems of bi-cofactor form. J. Phys. A Math. Gen. 35, 2947 (2002)MathSciNetCrossRefADSGoogle Scholar
  41. 191.
    Marciniak, K, Błaszak, M.: Non-Hamiltonian systems separable by Hamilton–Jacobi method. J. Geom. Phys. 58, 557 (2008)MathSciNetCrossRefADSGoogle Scholar
  42. 192.
    Marciniak, K., Błaszak, M.: Flat coordinates of flat Stäckel systems. Appl. Math. Comput. 268, 706 (2015)MathSciNetzbMATHGoogle Scholar
  43. 194.
    Marciniak, K., Rauch-Wojciechowski, S.: Two families of nonstandard Poisson structures for Newton equations. J. Math. Phys. 39, 6366 (1998)MathSciNetCrossRefGoogle Scholar
  44. 203.
    Morosi, C., Tondo, G.: Quasi-bi-Hamiltonian systems and separability. J. Phys. A Math. Gen. 30, 2799 (1997)MathSciNetCrossRefADSGoogle Scholar
  45. 204.
    Moser, J.: Various aspects of integrable Hamiltonian systems. Prog. Math. 8, 23 (1980)MathSciNetzbMATHGoogle Scholar
  46. 216.
    Panasyuk, A.: Veronese webs for bihamiltonian structures of higher corank. In: Urbański, P., Grabowski, J. (eds.), Poisson Geometry (Warsaw 1998). Banach Center Publications 51, Polish Academy of Sciences, Warsaw (2000)Google Scholar
  47. 217.
    Panasyuk, A.: Compatible lie brackets: Towards a classifcation. J. Lie Theory 24, 561 (2014)MathSciNetzbMATHGoogle Scholar
  48. 218.
    Pedroni, M.: Bi-Hamiltonian aspects of the separability of the Neumann system. Theor. Math. Phys. 133, 1722 (2002)MathSciNetCrossRefGoogle Scholar
  49. 223.
    Ratiu, P.S.: The C. Neumann problem as a completely integrable system on an adjoint orbit of a Lie algebra. Trans. Amer. Math. Soc. 264, 321 (1981)MathSciNetCrossRefGoogle Scholar
  50. 224.
    Rauch-Wojciechowski, S.: A bi-Hamiltonian formulation for separable potentials and its application to the Kepler problem and the Euler problem of two centers of gravitation. Phys. Lett. A 160, 149 (1991)MathSciNetCrossRefADSGoogle Scholar
  51. 225.
    Rauch-Wojciechowski, S., Marciniak, K., Lundmark, H.: Quasi-Lagrangian systems of Newton equations. J. Math. Phys. 40, 6366 (1999)MathSciNetCrossRefADSGoogle Scholar
  52. 232.
    Shabat, A.B.: The infinite-dimensional dressing dynamical system. Inverse Prob. 6, 303 (1992)MathSciNetCrossRefADSGoogle Scholar
  53. 233.
    Sklyanin, E.: Separation of variables in the Gaudin model. J. Sov. Math. 47, 2473 (1989)MathSciNetCrossRefGoogle Scholar
  54. 235.
    Sklyanin, E: Separation of variables. New trends. Prog. Theor. Phys. Suppl. 118, 35 (1995)MathSciNetCrossRefADSGoogle Scholar
  55. 247.
    Ten Eikelder, H.M.M.: On the local structure of recursion operators. Proc. Kon. Ned. Akad. A 89, 386 (1986)Google Scholar
  56. 250.
    Turiel, F.J.: Classification locale d’un couple de formes symplectiques Poisson-compatibles. C.R. Acad. Sci. Paris Ser. I Math. 308, 575 (1989)Google Scholar
  57. 253.
    Veselov, A., Shabat, A.B.: Dressing chain and spectral theory of Schrödinger operator. Funktsional. Anal. i Prilozhen. 27, 1 (1993)MathSciNetCrossRefGoogle Scholar
  58. 264.
    Wojciechowski, S.: Review of the recent results on integrability of natural Hamiltonian systems. In: Systèmes dynamiques non linéaires: intégrabilité et comportement qualitatif, pp. 294–327. Sém. Math. Sup., 102. Presses Univ. Montréal, Montreal, QC (1986)Google Scholar
  59. 272.
    Zeng, Y., Ma, W.X.: Separation of variables for soliton equations via their binary constrained flows. J. Math. Phys. 40, 6526 (1999)MathSciNetCrossRefADSGoogle Scholar
  60. 273.
    Zeng, Y., Ma, W.X.: The construction of canonical separated variables for binary constrained AKNS flow. Physica A 274, 505 (1999)CrossRefADSGoogle Scholar

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Authors and Affiliations

  • Maciej Błaszak
    • 1
  1. 1.Division of Mathematical PhysicsFaculty of Physics UAMPoznańPoland

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