Linearizing Completely Integrable Systems on Complex Algebraic Tori

  • P. van Moerbeke
Conference paper
Part of the NATO ASI Series book series (volume 37)


How does one recognize whether a Hamiltonian system is completely integrable ? Granted a system is integrable, how does one effectively integrate the problem ? The answer to these questions is unknown, up to this day! The proof of the Liouville theorem concerning integrals in involution and invariant tori is non-constructive : neither does it enable you to decide about its integrability, nor does it provide means for integrating the problem. Historically, the solutions to the classical systems that we know have required the most ingenious tricks by the best mathematical minds and they are only distinguished by their variety. The resolution of the Korteweg-de Vries equation by inverse spectral methods, some twenty years ago, has led to a number of Lie theoretical and algebraic geometrical methods for producing ordinary and partial differential equations, some of which are interesting from the point of view of mechanics and physics. They all give rise to so-called algebraically completely integrable systems; this stringent, but yet quite typical notion of integrability will be defined in § 2, whereas § 1 deals with the simple but fundamental example of Euler’s rigid body rotating about its center of mass.


Manifold Cond 


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  1. [1]
    Adler, M. and van Moerbeke, P.; 1) Completely integrale systems, Euclidean Lie algebras and curves, 2) Linearization of Hamiltonian systems, Jacobi varieties and Representation theory; Adv. in Math., 38, pp. 267–317, 318–379, (1980).Google Scholar
  2. [2]
    Adler, M. and van Moerbeke, P.; The algebraic integrability of geodesic flow on SO(4), Invent. Math., 67, pp. 297–326, (1982) with an appendix by D. MumfordGoogle Scholar
  3. [3]
    Adler, M. and van Moerbeke, P.; Kowalewski’s asymptotic method, Kac-Moody Lie algebras and regularization, Comm. Math. Phys., 83, pp. 83–106, (1982).MATHCrossRefGoogle Scholar
  4. [4]
    Adler, M. and van Moerbeke, P.; Geodesic flow on SO(4) and the intersection of quadrics, Proc. Natl. Acad. Sci. USA, 81, pp. 4613–4616, (1984).MATHCrossRefGoogle Scholar
  5. [5]
    Adler, M. and van Moerbeke, P.; A new integrable geodesic flow on SO(4). Probability, Statistical mechanics and number theory, Adv. in Math. Suppl. Studies, vol. 9, (1986).Google Scholar
  6. [6]
    Adler, M. and van Moerbeke, P.; A systematic Approach towards solving Integrable Systems, Perspectives in Mathematics, Academic Press (to appear in 1987 ).Google Scholar
  7. [7]
    Adler, M. and van Moerbeke, P.; A full classification of algebraically completely integrable geodesic flows on SO(4), (to appear in 1986 ).Google Scholar
  8. [8]
    Arnold, V.I.; Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg-Berlin, (1978).MATHGoogle Scholar
  9. [9]
    Griffiths, P.A.; Linearizing flows and a cohomological interpretation of Lax equations, Amer. J. of Math., 107, pp. 1445–1483, (1985).MATHGoogle Scholar
  10. [10]
    Griffiths, P. and Harris, J.; Principles of algebraic geometry. New York: Wiley-Interscience, (1978).MATHGoogle Scholar
  11. [11]
    Haine, L.; Geodesic flow on SO(4) and Abelian surfaces, Math. Ann., 263, pp. 435–472, (1983).MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Haine, L.; The algebraic complete integrability of geodesic flow on SO(N), Comm. Math. Phys., 94, pp. 271287, (1984).Google Scholar
  13. [13]
    Kowalewski, S.; Sur le problème de la rotation d’un corps solide autour d’un point fixe, Acta Math., 12, pp. 177–232, (1889).MathSciNetCrossRefGoogle Scholar
  14. [14]
    Kowalewski, S.; Sur une propriété du système d’équations différentielles qui définit la rotation d’un corps solide autour d’un point fixe, Acta Math., 14, pp. 81–93, (1889).CrossRefGoogle Scholar
  15. [15]
    Kozlov, V.V.; Integrability and non-integrability in Hamiltonian mechanics, Uspekhi Mat. Nauk,38: 1, pp. 3–67, (1983), (Transl.:Russian Math. Surveys, 38: 1, pp. 1–76, (1983)).Google Scholar
  16. [16]
    van Moerbeke, P.; The complete integrability of Hamiltonian system, Proceedings of the EQUADIFF conference, Würzburg (August 1982), Springer-Verlag Lecture Notes, 1017, pp. 462–475.Google Scholar
  17. [17]
    van Moerbeke, P.; Algebraic geometrical methods in Hamiltonian mechanics, Phil. Trans. Royal Society London, A315, pp. 379–390, (1985), ( Royal Society Meeting, Nov. 1984 ).Google Scholar
  18. [18]
    van Moerbeke, P.; Algebraic complete integrability of Hamiltonian systems and Kac-Moody Lie algebras, Proc. Int. Congr. of Math., Warszawa, August 1983.Google Scholar
  19. [19]
    Perelomov, A.M.; Some remarks on the integrability of the equations of motion of a rigid body in an ideal fluid, Funct. Anal. Appl.,15, pp. 83–85, (1981), transi. 144–146.Google Scholar
  20. [20]
    Reiman A.,Semenov-Tian-Shansky, M.; A new integrable case of the motion of the 4-dimensional rigid body, Comm. Math. Phys., 105, pp. 461–472, (1986).MathSciNetMATHGoogle Scholar
  21. [21]
    Yoshida, H.; Necessary conditions for the existence of algebraic first integrals. I. Kowalewski’s Exponents. II. Conditions for algebraic integrability, Celestial Mech., 31, pp. 363–379, 381–399, (1983).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • P. van Moerbeke
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of LouvainLouvain-la-NeuveBelgium
  2. 2.Brandeis UniversityWalthamUSA

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