Advertisement

Regular and Chaotic Dynamics

, Volume 17, Issue 5, pp 431–438 | Cite as

On the cases of Kirchhoff and Chaplygin of the Kirchhoff equations

Article

Abstract

It is proven that the completely integrable general Kirchhoff case of the Kirchhoff equations for B ≠ 0 is not an algebraic complete integrable system. Similar analytic behavior of the general solution of the Chaplygin case is detected. Four-dimensional analogues of the Kirchhoff and the Chaplygin cases are defined on e(4) with the standard Lie-Poisson bracket.

Keywords

Kirchhoff equations Kirchhoff case Chaplygin case algebraic integrable systems 

MSC2010 numbers

70E40 70E45 70E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, M. and van Moerbeke, P., The Complex Geometry of the Kowalewski-Painlevé Analysis, Invent. Math., 1989, vol. 97, no. 1, pp.3–51.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Adler, M., van Moerbeke, P., and Vanhaecke, P., Algebraic Integrability, Painlevé Geometry and Lie Algebras, Ergeb. Math. Grenzgeb. (3), vol. 47, Berlin: Springer, 2004.MATHGoogle Scholar
  3. 3.
    Borisov, A. V., Necessary and Sufficient Conditions of Kirchhoff Equation Integrability, Regul. Chaot. Dyn., 1996, vol. 1, no. 2, pp. 61–76 (Russian).MATHGoogle Scholar
  4. 4.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2001 (Russian).MATHGoogle Scholar
  5. 5.
    Chaplygin, S.A., Selected Works, Moscow: Nauka, 1976 (Russian).Google Scholar
  6. 6.
    Dragović, V. and Gajić, B., An L-A Pair for the Hess-Appel’rot System and a New Integrable Case for the Euler-Poisson Equations on so(4) × so(4), Proc. Roy. Soc. Edinburgh Sect. A, 2001, vol. 131, no. 4, pp. 845–855.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dragović, V. and Gajić, B., The Lagrange Bitop on so(4) × so(4) and Geometry of Prym Varieties, Amer. J. Math., 2004, vol. 126, no. 5, pp. 981–1004.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dragović, V. and Gajić, B., Systems of Hess-Appel’rot Type, Comm. Math. Phys., 2006, vol. 265, no. 2, pp. 397–435.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dragović, V., Gajić, B., and Jovanović, B., Systems of Hess-Appel’rot Type and Zhukovskii Property, Int. J. Geom. Methods Mod. Phys., 2009, vol. 6, no. 8, pp. 1253–1304.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Golubev, V. V., Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Moscow: Gostekhizdat, 1953 [English transl.: Jerusalem, Israel Program for Scientific Translations, 1960; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington].Google Scholar
  11. 11.
    Kirchhoff, G.R., Vorlesungen über mathematische Physics: Bd. 1: Mechanik, Leipzig: Teubner, 1874.Google Scholar
  12. 12.
    Kowalevski, S., Sur le problème de la rotation d’un corps solide autour d’un point fixe, Acta Math., 1889, vol. 12, no. 2, pp. 177–232.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kozlov, V.V., Methods of Qualitative Analysis in the Dynamics of a Rigid Body, Moscow: Moskov. Gos. Univ., 1980 (Russian).MATHGoogle Scholar
  14. 14.
    Kozlov, V.V. and Onischenko, D.A., Nonintegrability of Kirchhoff Equations, Dokl. Akad. Nauk SSSR, 1982, vol. 266, no. 6, pp. 1298–1300 [Soviet Math. Dokl., 1982, vol. 26, no. 2, pp. 495–498].MathSciNetGoogle Scholar
  15. 15.
    Lamb, H., Hydrodynamics, 6th ed., New York: Dover, 1945.Google Scholar
  16. 16.
    Lesfari, A., Prym Varieties and Applications, J. Geom. Phys., 2008, vol. 58, no. 9, pp. 1063–1079.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Perelomov, A. M., Some Remarks on the Integrability of the Equations of Motion of a Rigid Body in an Ideal Fluid, Funktsional. Anal. i Prilozhen., 1981, vol. 15, no. 2, pp. 83–85 [Funct. Anal. Appl., 1981, vol. 15, no. 2, pp. 144–146].MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rubanovskiy, V. N., Applications of Method of Small Parameter in Equations of Rigid Body in Fluid, Vestn. Mosk. Univ., Ser. 1. Mat. Mekh., 1967, no. 3, pp. 80–86 (Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Mathematical InstituteSerbian Academy of Science and ArtBelgradeSerbia
  2. 2.Group of Mathematical PhysicsUniversity of LisbonLisbonPortugal

Personalised recommendations