The Chaotic Motion of a Rigid Body Rotating About a Fixed Point

  • F. El-Sabaa
  • M. El-Tarazi
Part of the NATO ASI Series book series (NSSB, volume 272)


The dynamical system with two degrees of freedom has been studied by many authors. It is well known, that if the system possesses a first integral besides the integral of energy, then the system is completely integrable. After the fundamental works of Kolmogorov [1], Arnold [2] and Moser [3] (KAM) in 1960–1968, many theoretical and numerical results were presented by authors who treated the problem, when the system contained only the integral of energy (Jacobi’s integral). The results of (kam) have clarified the picture of nonintegrable systems through the small perturbations of integrable systems; for small perturbations we get very regular orbits, lying apparently on invariant tori, while for larger perturbations a part of the tori seems to be destroyed, and erratic orbits appear instead, filling the so-called stochastic region.


Periodic Solution Periodic Orbit Rigid Body Periodic Motion Eulerian Angle 
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.
    A Kolmogorov, On Conservation of Conditionally periodic motions under small perturbations of the Hamiltonian, Dokl. Akad. Nauk SSSR. 98: 527 (1954)MathSciNetMATHGoogle Scholar
  2. 2.
    V. Arnold, Small divisor problems in classical and celestial mechanics, Russian Mathematical Survey. 18: 581 (1963)Google Scholar
  3. 3.
    J. Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton (1973)Google Scholar
  4. 4.
    L. Euler, Decouverte d’une nouveau principle de mechanique, Memoires del’Academie des Sciences de Berlin, 14: 154 (1758)Google Scholar
  5. 5.
    A. Deprit, Free rotation of a rigid body studied in the phase plane. American Journal of Physics. 35: 224 (1967)CrossRefGoogle Scholar
  6. 6.
    Iu. Barkin, E. levlev, Periodic motion of a rigid body with a fixed point in the gravity field of two centers, Prikladnia Matemati e Mekhanik. 41: 574 (1977)Google Scholar
  7. 7.
    V. Demin, F. Kiselev, On periodic motions of a rigid body in a central Newtonian field, Prikladnia Matematik e Mekhanik.38: 224 (1974)Google Scholar
  8. 8.
    F. El-Sabaa, The periodic solution of a rigid body in a central Newtonian field, Astrophysics and Space Science, 162: 235 (1989)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    J. Lagrange, Mechanique analytique, Gauthier-Villars, Paris (1888)Google Scholar
  10. 10.
    G. Applerot, 1893. Some additions of paper N. Delone “algebraic integrals of a heavy rigid body about fixed point”. Troda Otdelenii Fizicheskikh Obshchestve Liubitelei Estesvoznaiia. G: 1 (1893)Google Scholar
  11. 11.
    V. Kozlov, Qualitative analysis method in dynamics of a rigid body, Moscow State University Press, Moscow (1980)Google Scholar
  12. 12.
    F. El-Sabaa. About the periodic solution of Kovaleveskaya’s top by using Liapunov’s method. Journal of the University of Kuwait (Science) 16; 21 (1989)MathSciNetMATHGoogle Scholar
  13. 13.
    E. Kharlamov. On the equations of motion for a heavy body with fixed point. Priklad Matematik e Mekhanik. 27:1070 (1963)MathSciNetMATHGoogle Scholar
  14. 14.
    A Liapunov. Stability of motion. Academic Press, New York (1966)Google Scholar
  15. 15.
    M. Hénon, C. Heiles, The applicability of the third integral of motion: some numerical experiments. Astronomical Journal 69:73 (1964)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    H. Poincaré, Methodes Nouvelles de la Mecanique Celeste. Dover, New York (1957)MATHGoogle Scholar
  17. 17.
    G. Birkhoff. Dynamical systems. American Mathematical Society New York (1927)MATHGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • F. El-Sabaa
    • 1
  • M. El-Tarazi
    • 2
  1. 1.Department of Mathematics, Faculty of EducationAin Shamis UniversityCairoEgypt
  2. 2.Department of MathematicsKuwait UniversityKuwait

Personalised recommendations