Advertisement

Celestial Mechanics and Dynamical Astronomy

, Volume 58, Issue 4, pp 387–391 | Cite as

Integrability of the Yang-Mills Hamiltonian system

  • S. Kasperczuk
Article

Abstract

This paper considers the integrability of generalized Yang-Mills system with the HamiltonianH a (p, q)=1/2(p 1 2 +p 2 2 +a1q 1 2 +a2q 2 2 )+1/4q 1 4 +1/4a3q 2 4 + 1/2a4q 1 2 q 2 2 . We prove that the system is integrable for the cases: (A)a1=a2,a3=a4=1; (b)a1=a2,a3=1,a4=3; (C)a1=a2/4,a3=16,a4=6. Our main result is the presentation of these integrals. Only for cases A and B does the Yang-Mills Hamiltonian possess the Painlevé property. Therefore the Painlevé test does not take account of the integrability for the case C.

Key words

Hamiltonian systems Painlevé test integrability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abraham, R. and Marsden, J.: 1978,Foundations of Mechanics, 2nd ed., Benjamin/Cumings, Reading Massachusetts.Google Scholar
  2. Arnold, V.I.: 1978,Mathematical Methods of Classical Mechanics, Gradue Texts in Mathematics, Vol. 60, Springer-Verlag, N.Y.Google Scholar
  3. Bountis, T., Segur, H. and Vivaldi, F.: 1982, ‘Integrable Hamiltonian Systems and the Painlevé Property’,Phys. Rev. 25A, 1257–1264.Google Scholar
  4. Dorizzi, B., Grammaticos, B. and Ramani, A.: 1983, ‘A New Class of Integrable Systems’,J. Math. Phys. 24, 2282–2288.Google Scholar
  5. Ercolani, N. and Siggia, E.D.: 1986, ‘Painlevé Property and Integrability’,Phys. Lett. 119A, 112–116.Google Scholar
  6. Fridberg, R., Lee, T.D. and Sirlin, A.: 1976, ‘Class of Scalar-Field Solutions in Three Space Dimensions’,Phys. Rev. 13D, 2739–2761.Google Scholar
  7. Grammaticos, B., Dorizzi, B. and Padjen, R.: 1982, ‘Painlevé Property and Integrals of Motion for the Henon-Heiles System’,Phys. Lett. 89A, 111–113.Google Scholar
  8. Kowalevski, S.: 1889, ‘Sur le Problème de la Rotation d'un Corps Solide Autour d'un Point Fixe’,Acta Math. 12, 177–232.Google Scholar
  9. Kowalevski, S.: 1890, ‘Sur une Propriété du Système d'Équations Différentielles qui Définit la Rotation d'un Corps Solide Autour d'un Point Fixe’,Acta Math. 14, 81–93.Google Scholar
  10. Newell, A.C., Tabor, M. and Zeng, Y.B.: 1987, ‘A Unified Approach to Painlevé Expressions’,Physica D 29, 1–68.Google Scholar
  11. Rajaraman, R. and Weinberg, E.J.: 1975, ‘Internal Symmetry and the Semi- Classical Method in Quantum Field Theory’,Phys. Rev. D 11, 2950–2966.Google Scholar
  12. Yoshida, H.: 1986, ‘Existence of Exponentially Unstable Periodic Solutions and the Non-Integrability of Homogeneous Hamiltonian’,Physica D 21, 163–170.Google Scholar
  13. Zakharov, V.E., Ivanov, M.F. and Shoor, L.I.: 1979, ‘On Anomalously Slow Stochastization in Certain Two-Dimensional Models of Field Theory’,Zh. Eksp. Teor. Fiz. Lett. 30, 39–44.Google Scholar
  14. Ziglin, S.L.: 1983, ‘Branching of Solutions and Non-Existence of First Integrals in Hamiltonian Mechanics’,Functional Anal. Appl. 17, 6–17.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • S. Kasperczuk
    • 1
  1. 1.Institute of PhysicsPedagogical UniversityZielona GóraPoland

Personalised recommendations