Non-Integrability Criterion of Hamiltonian Systems based on Ziglin’s Theorem and its Relation to the Singular Point Analysis

  • Haruo Yoshida
Part of the NATO ASI Series book series (NSSB, volume 331)


After two examples of the singular point analysis, a sufficient condition (criterion) for the non-existence of an additional analytic integral is given for n-degree-of-freedom Hamiltonian systems with a homogeneous potential, which justifies the validity of the singular point analysis. This criterion is based on Ziglin’s theorem, which will be reviewed extensively from the basic ideas.


Periodic Solution Hamiltonian System Characteristic Exponent Monodromy Matrix Jacobi Elliptic Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Block, H., 1908, Arkiv för Mat. Astron. Fys. 5, No. 9, 1.Google Scholar
  2. [2]
    Kowalevski, S., 1890, Acta Math. 14, 81.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    Ramani, A., Grammaticos, B. and Bountis, T., 1989, Phys. Rep. 180, 159.CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    Siegel, C. L., 1941, Ann. Math. 42, 127.CrossRefGoogle Scholar
  5. [5]
    Yoshida, H., 1983, Celestial Mech. 31, 363.CrossRefzbMATHMathSciNetADSGoogle Scholar
  6. [6]
    Yoshida, H., 1986, Physica D 21, 163.CrossRefzbMATHMathSciNetADSGoogle Scholar
  7. [7]
    Yoshida, H., 1987, Physica D 29, 128.CrossRefzbMATHMathSciNetADSGoogle Scholar
  8. [8]
    Yoshida, H., 1988, ‘Ziglin analysis for proving non-integrability of Hamiltonian system’, in Finite Dimensional Integrable Nonlinear Dynamical Systems, P.G.L. Leach and W.H. Steeb, eds. (World Scientific, Singapore) 74.Google Scholar
  9. [9]
    Yoshida, H., 1989, Phys. Lett. A 141, 108.CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    Ziglin, S.L., 1983, Functional Anal. Appl. 16, 181.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Haruo Yoshida
    • 1
  1. 1.National Astronomical ObservatoryMitaka, Tokyo 181Japan

Personalised recommendations