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An Elementary approach to Integrability Condition for the Euler Equations on Lie Algebra so(4)

  • Sasho Ivanov Popov
  • Jean-Marie Strelcyn
Part of the NATO ASI Series book series (NSSB, volume 331)

Abstract

The completely elementary derivation of the Manakov integrability condition of the Euler equations on the Lie algebra so(4) is given. Four functionally independent first integrals are written explicitly for all values of parameters. Some simple cases of partial integrability are found.

Keywords

Euler Equation Invariant Manifold Nonlinear Evolution Equation Geodesic Flow Bulgarian Academy 
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References

  1. 1.
    M.J. Ablowitz, A. Ramani, H. Segur, Nonlinear Evolution Equations and Ordinary Differential Equations of Painleve type, Lett. Nuov. Cim., 23, 333–338 (1978).CrossRefMathSciNetGoogle Scholar
  2. 2.
    M.J. Ablowitz, A. Ramani, H. Segur, A Connection Between Nonlinear Evolution Equations and Ordinary Differential Equations of P-type I, J. Math. Phys., 21, 715–721 (1980).CrossRefzbMATHMathSciNetADSGoogle Scholar
  3. 3.
    M. Adler and P. van Moerbeke, The algebraic integrability of geodesic flow on SO(4), Invent. Math., 67, 297–331 (1982).CrossRefzbMATHMathSciNetADSGoogle Scholar
  4. 4.
    Yu.A. Arkhangelskii, Analytical dynamics of the rigid body (in Russian), Nauka, Moscow (1977).Google Scholar
  5. 5.
    V.I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Math. 60, 2nd edition, Springer-Verlag, Berlin (1989).CrossRefGoogle Scholar
  6. 6.
    O.I. Bogoyavlensky, Integrable Euler Equations on Six-Dimensional Lie Algebras, Doklady Akad. Nauk USSR, (in Russian) 268, 11–15 (1983). English transl. Soviet Math. Doklady, 27, Nl, 1-5 (1983).Google Scholar
  7. 7.
    A.T. Fomenko, Integrability and nonintegrability in geometry and mechanics, Do-drecht: Kluwer Academic Publishers (1988).CrossRefzbMATHGoogle Scholar
  8. 8.
    L. Haine, Geodesic flow on SO(4) and abelian surfaces, Math. Ann., 263, 435–472 (1983).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    L. Haine, The algebraic complete integrability of geodesic flow on SO(N), Commun. Math. Phys., 94, 271–287 (1984).CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    S.V. Kovalevskaya, Sur le probleme de la rotation d’un corps solide autour d’un point fixe, Acta Math., 12, 177–232 (1889).CrossRefMathSciNetGoogle Scholar
  11. 11.
    S.V. Kovalevskaya, Sur une propriete du systeme d’equations differentielles qui definit la rotation d’un corps solide autour d’un point fixe, Acta Math., 14, 81–93 (1880).MathSciNetGoogle Scholar
  12. 12.
    P. J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Math. 107, Springer-Verlag, Berlin (1986).CrossRefzbMATHGoogle Scholar
  13. 13.
    A.M. Perelomov, Integrable systems of classical mechanics and Lie algebras. The motion of rigid body with fixed point (in Russian), Preprints of Inst. of Theor. Exper. Phys., Moscow 147 (1983).Google Scholar
  14. 14.
    A.M. Perelomov, Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhauser Verlag, Basel (1990).CrossRefGoogle Scholar
  15. 15.
    S.I. Popov, On the existence of depending on p, q, r, γ first integral of the Euler-Poisson system, Theoretical and Applied Mechanics. (Bulgarian Academy of Sciences), (in Bulgarian), 2, 28–33 (1981).Google Scholar
  16. 16.
    S.I. Popov, On the non-existence of a new first integral F(p, q, r, γ, γ′) = const of the problem of a heavy rigid body motion about a fixed point, Reports of Bulgarian Academy of Sciences, (in Russian), 38(5), 583–586 (1985).zbMATHGoogle Scholar
  17. 17.
    S.I. Popov, On the non-existence of a new first integral (p, q, r, γ, γ′) = const of the problem of a heavy rigid body motion about a fixed point, Theoretical and Applied Mechanics. (Bulgarian Academy of Sciences), (in Russian), 4, 17–23 (1988).Google Scholar
  18. 18.
    S.I. Popov, On the motion of a heavy rigid body about a fixed point, Acta Mechanica, 85, 1–11 (1990).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Sasho Ivanov Popov
    • 1
  • Jean-Marie Strelcyn
    • 2
    • 3
    • 4
  1. 1.Institute of Metal ScienceBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Departement de MathematiquesUniversite de RouenMont Saint Aignan CedexFrance
  3. 3.Laboratoire AnalyseGeometrie et ApplicationsFrance
  4. 4.Departement de MathematiquesInstitut GalileeVilletaneuseFrance

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