Acta Applicandae Mathematica

, Volume 87, Issue 1–3, pp 93–121 | Cite as

Superintegrable Hamiltonian Systems: Geometry and Perturbations

  • Francesco Fassò


Many and important integrable Hamiltonian systems are ‘superintegrable’, in the sense that there is an open subset of their 2d-dimensional phase space in which all motions are linear on tori of dimension n<d. A thorough comprehension of these systems requires a description which goes beyond the standard notion of Liouville–Arnold integrability, that is, the existence of an invariant fibration by Lagrangian tori. Instead, the natural object to look at is formed by both the fibration by the (isotropic) invariant tori and by its (coisotropic) polar foliation, which together form what in symplectic geometry is called a ‘dual pair’, or ‘bifoliation’, or ‘bifibration’. We review this geometric structure, relating it to the dynamical properties of superintegrable systems and pointing out its importance for a thorough understanding of these systems.


superintegrability dual pairs Hamiltonian perturbation theory rigid body 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, R., Marsden, J. E. and Ratiu, T.: Manifolds, Tensor Analysis, and Applications, Springer, New York, 1993. Google Scholar
  2. 2.
    Arnold, V. I.: Mathematical Methods of Classical Mechanics, 2nd edn, Springer, New York, 1989. Google Scholar
  3. 3.
    Bates, L. M.: Examples for obstructions to action-angle coordinates, Proc. Roy. Soc. Edinburgh A 110 (1988), 27–30. Google Scholar
  4. 4.
    Bates, L. M.: Monodromy in the champagne bottle, J. Appl. Math. Phys. (ZAMP) 42 (1991), 837–847. CrossRefGoogle Scholar
  5. 5.
    Benettin, G.: The elements of Hamiltonian perturbation theory, to appear in the Proceedings of the School on Hamiltonian Systems and Fourier Analysis held in Porquerolles, September 2001. Google Scholar
  6. 6.
    Benettin, G., Cherubini, A. M. and Fassò, F.: Regular and chaotic motions of the fast rotating rigid body: A numerical study, Discrete Contin. Dynam. Systems, Ser. B 2 (2002), 521–540. Google Scholar
  7. 7.
    Benettin, G. and Fassò, F.: Fast rotations of the rigid body: A study by Hamiltonian perturbation theory. Part I, Nonlinearity 9 (1996), 137–186. CrossRefGoogle Scholar
  8. 8.
    Benettin, G., Fassò, F. and Guzzo, M.: Long-term stability of proper rotations of the Euler perturbed rigid body, Comm. Math. Phys. 250 (2004), 133–160. Google Scholar
  9. 9.
    Blaom, A. D.: A geometric setting for Hamiltonian perturbation theory, Mem. Amer. Math. Soc. 153 (2001), 1–112. Google Scholar
  10. 10.
    Bogoyavlenskij, O. I.: Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys. 196 (1998), 19–51. Google Scholar
  11. 11.
    Bolsinov, A. V. and Jovanovich, B.: Non-commutative integrability, moment map and geodesic flows, Ann. Global Anal. Geom. 23 (2003), 305–322. Google Scholar
  12. 12.
    Born, M.: The Mechanics of the Atom, Frederick Ungar Publishing, New York, 1960. Google Scholar
  13. 13.
    Cushman, R. H.: Normal form for Hamiltonian vectorfields with periodic flow, In: S. Sternberg (ed.), Differential Geometric Methods in Mathematical Physics, Reidel, Dordrecht, 1984, pp. 125–144. Google Scholar
  14. 14.
    Cushman, R. and Bates, L.: Global Aspects of Classical Integrable Systems, Birkhäuser, Basel, 1997. Google Scholar
  15. 15.
    Cushman, R. H., Dullin, H. R., Giacobbe, A., Holm, D. D., Joyeaux, M., Linch, P., Sadovskií, D. A. and Zhilinskií, B. I.: CO2 molecule as a quantum realization of the 1:1:2 resonant swing spring with monodromy, Phys. Rev. Lett. 93 (2004), 024302. PubMedGoogle Scholar
  16. 16.
    Dazord, P. and Delzant, T.: Le probleme general des variables actions-angles, J. Diff. Geom. 26 (1987), 223–251. Google Scholar
  17. 17.
    Duistermaat, J. J.: On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687–706. Google Scholar
  18. 18.
    Dullin, H. and Hanßmann, H.: The degenerate C. Neumann system. I: Symmetry reduction and convexity, Preprint, 2004. Google Scholar
  19. 19.
    Evans, N. W.: Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666–5676. PubMedGoogle Scholar
  20. 20.
    Fassò, F.: Hamiltonian perturbation theory on a manifold, Cel. Mech. Dyn. Astr. 62 (1995), 43–69. Google Scholar
  21. 21.
    Fassò, F.: The Euler–Poinsot top: A non-commutatively integrable system without global action-angle coordinates, J. Appl. Math. Phys. (ZAMP) 47 (1996), 953–976. Google Scholar
  22. 22.
    Fassò, F.: Lectures on integrable Hamiltonian systems, Unpublished lecture notes available from the author, 1996. Google Scholar
  23. 23.
    Fassò, F.: Quasi periodicity of motions and complete integrability of Hamiltonian systems, Ergodic Theory Dynam. Systems 18 (1998), 1349–1362. Google Scholar
  24. 24.
    Fassò, F. and Ratiu, T.: Compatibility of symplectic structures adapted to noncommutatively integrable systems, J. Geom. Phys. 27 (1998), 199–220. Google Scholar
  25. 25.
    Fassò, F. and Giacobbe, A.: Geometric structure of “broadly integrable” Hamiltonian systems, J. Geom. Phys. 44 (2002), 156–170. Google Scholar
  26. 26.
    Fassò, F., Giacobbe, A. and Sansonetto, N.: Periodic flows, Poisson structures, and nonholonomi mechanics, Preprint, 2004. Google Scholar
  27. 27.
    Giacobbe, A.: Some remarks on the Gelfand–Cetlin system, J. Phys. A 35 (2002), 10591–10605. Google Scholar
  28. 28.
    Hanßmann, H.: Quasi-periodic motions of a rigid body. I: Quadratic Hamiltonians on the sphere with a distinguished parameter, Regul. Chaotic Dynam. 2 (1997), 41–57. Google Scholar
  29. 29.
    Karasev, M. V. and Maslov, V. P.: Nonlinear Poisson Brackets. Geometry and Quantization, Translations of the Amer. Math. Soc., Vol. 119, Amer. Math. Sci., Providence, RI, 1993. Google Scholar
  30. 30.
    Kibler, M. and Winternitz, P.: Periodicity and quasi-periodicity for super-integrable Hamiltonian systems, Phys. Lett. A 147 (1990), 338–342. Google Scholar
  31. 31.
    Libermann, P. and Marle, C.-M.: Symplectic Geometry and Analytical Mechanics, D. Reidel, Dordrecht, 1987. Google Scholar
  32. 32.
    Lochak, P.: Canonical perturbation theory via simultaneous approximation, Russian Math. Surveys 47 (1992), 57–133. Google Scholar
  33. 33.
    Marsden, J. E. and Ratiu, T. S.: Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Springer, New York, 1994. Google Scholar
  34. 34.
    Mazzocco, M.: KAM theory for generic analytic perturbations of the Euler system, Z. Angew. Math. Phys. 48 (1997), 193–219. Google Scholar
  35. 35.
    Meigniez, G.: Submersion, fibrations and bundles, Trans. Amer. Math. Soc. 354 (2002), 3771–3787. Google Scholar
  36. 36.
    Mischenko, A. S. and Fomenko, A. T.: Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978), 113–121. Google Scholar
  37. 37.
    Moser, J.: Regularization of Kepler’s problem and the averaging method on a manifold, Comm. Pure Appl. Math. 23 (1970), 609–636. Google Scholar
  38. 38.
    Nekhoroshev, N. N.: Action-angle variables and their generalizations, Trans. Moskow Math. Soc. 26 (1972), 180–198. Google Scholar
  39. 39.
    Nekhoroshev, N. N.: An exponential estimate of the time of stability of nearly integrable hamiltonian systems, Russian Math. Surveys 32 (1977), 1–65. Google Scholar
  40. 40.
    Parasyuk, I. O.: Co-isotropic invariant tori of Hamiltonian systems in the quasiclassical theory of the motion of a conduction electron, Ukrainian Math. J. 42 (1990), 308–312. Google Scholar
  41. 41.
    Parasyuk, I. O.: Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori, Ukrainian Math. J. 45 (1993), 85–93. Google Scholar
  42. 42.
    Parasyuk, I. O.: Reduction and coisotropic invariant tori of Hamiltonian systems with non-Poisson commutative symmetries. II, Ukrainian Math. J. 46 (1994), 991–1002. Google Scholar
  43. 43.
    Pöschel, J.: Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z. 213 (1993), 187–216. Google Scholar
  44. 44.
    Steenrod, N.: The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, 1951. Google Scholar
  45. 45.
    Tempesta, P., Winternitz, P., Harnad, J., Miller, W., Pogosyan, G. and Rodriguez, M.: Super-integrability in Classical and Quantum Systems, CRM Proceedings & Lecture Notes 37, Amer. Math. Soc., 2004. Google Scholar
  46. 46.
    Vũ Ngoc, S.: Bohr–Sommerfeld conditions for integrable systems with critical manifolds of focus–focus type, Comm. Pure Appl. Math. 53 (2003), 143–217. Google Scholar
  47. 47.
    Weinstein, A.: The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), 523–557. Google Scholar
  48. 48.
    Woodhouse, N. M. J.: Geometric Quantization, 2nd edn, Clarendon Press, Oxford, 1994. Google Scholar
  49. 49.
    Zung, N. T.: Torus actions and integrable systems, Preprint, 2003. Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Università di Padova, Dipartimento di Matematica Pura e ApplicataPadovaItaly

Personalised recommendations