Integrable and Superintegrable Systems

  • José F. CariñenaEmail author
  • Alberto Ibort
  • Giuseppe Marmo
  • Giuseppe Morandi


There is no generally accepted definition of integrability that would include the various instances which are usually associated with the word “integrable". Occasionally the word ‘solvable’ is also used more or less as synonymous, but to emphasize the fact that the system need not be Hamiltonian.


Vector Field Poisson Structure Hamiltonian Vector Field Superintegrable System Double Vector Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [MM74]
    Markus, L., Meyer, K.R.: Generic Hamiltonian dynamical systems are neither integrable nor ergodic. Mem. Am. Math. Soc. 144, 1–52 (1974)MathSciNetGoogle Scholar
  2. [IM12]
    Ibort, A., Marmo, G., Rodríguez, M.A., Tempesta, P.: Nilpotent Integrability (Preprint) (2012).Google Scholar
  3. [Kl72]
    Klein, F.: A comparative review of recent researches in geometry. Univ. of Erlangen (1872)Google Scholar
  4. [Kl92]
    Klein, F.: A comparative review of recent researches in geometry. Bull. New York Math. Soc. 2, 215–249 (1892). arXiv: 0807.3161 v1[math.H0]
  5. [Ba86]
    Banyaga, A.: On isomorphic classical diffeomorphism groups I. Proc. Am. Math. Soc. 98, 113–118 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BG88]
    Batlle, C., Gomis, J., Pons, J.M., Román-Roy, N.: Lagrangian and Hamiltonian constraints for second-order singular Lagrangians. J. Phys. A: Math. Gen. 21, 2693–2703 (1988)CrossRefzbMATHADSGoogle Scholar
  7. [Gr00]
    Grabowski, J.: Isomorphisms of Poisson and Jacobi brackets. Poisson geometry. Banach Center Publ. 51, 79–85 (2000)MathSciNetGoogle Scholar
  8. [FL89]
    de Filippo, S., Landi, G., Marmo, G., Vilasi, G.: Tensor fields defining a tangent bundle structure. Ann. Inst. H. Poincaré 50, 205–218 (1989)zbMATHGoogle Scholar
  9. [Le34]
    Levi-Civita, T.: A general survey of the theory of adiabatic invariants. J. Math. Phys. 13, 18–40 (1934)Google Scholar
  10. [GM93]
    Giordano, M., Marmo, G., Rubano, C.: The inverse problem in the Hamiltonian formalism: integrability of linear Hamiltonian fields. Inverse Prob. 9, 443–467 (1993)MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. [EM10]
    Ercolessi, E., Marmo, G., Morandi, G.: From the equations of motion to the canonical commutation relations. Riv. Nuovo Cim. 33, 401–590 (2010)Google Scholar
  12. [At82]
    Atiyah, M.: Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14, 1–15 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [MM97]
    Man’ko, V.I., Marmo, G., Sudarshan, E.C.G., Zaccaria, F.: \(f\)-oscillators and nonlinear coherent states. Phys. Scripta 55, 528–541 (1997)CrossRefADSGoogle Scholar
  14. [LM97]
    López-Peña, R., Manko, V.I., Marmo, G.: Wigner problem for a precessing dipole. Phys. Rev. A 56, 1126–1130 (1997)CrossRefADSGoogle Scholar
  15. [Ne02]
    Nekhoroshev, N.N.: Generalizations of Gordon theorem. Regular Chaotic Dyn. 7, 239–247 (2002)MathSciNetCrossRefzbMATHADSGoogle Scholar
  16. [Go69]
    Gordon, W.B.: On the relation between period and energy in periodic dynamical systems. J. Math. Mech. 19, 111–114 (1969)MathSciNetzbMATHGoogle Scholar
  17. [DM05]
    D’Avanzo, A., Marmo, G.: Reduction and unfolding: the Kepler problem. Int. J. Geom. Meth. Mod. Phys. 2, 83–109 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [DM05b]
    D’Avanzo, A., Marmo, G., Valentino, A.: Reduction and unfolding for quantum systems: the hydrogen atom. Int. J. Geom. Meth. Mod. Phys. 2, 1043–1062 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [IM98]
    Ibort, A., Marmo, G.: A new look at completely integrable systems and double Lie groups. Contemp. Math. 219, 159–172 (1998)MathSciNetCrossRefGoogle Scholar
  20. [La68]
    Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. XXI, 467–90 (1968)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • José F. Cariñena
    • 1
    Email author
  • Alberto Ibort
    • 2
  • Giuseppe Marmo
    • 3
  • Giuseppe Morandi
    • 4
  1. 1.Departamento de Física TeóricaUniversidad de ZaragozaZaragozaSpain
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridMadridSpain
  3. 3.Dipartimento di FisicheUniversitá di Napoli “Federico II”NapoliItaly
  4. 4.INFN Sezione di BolognaUniversitá di BolognaBolognaItaly

Personalised recommendations