Nonintegrability, Separatrices Crossing and Homoclinic Orbits in the Problem of Rotational Motion of a Satellite

  • Andrzej J. Maciejewski
  • Krzysztof Goździewski
Part of the NATO ASI Series book series (NSSB, volume 298)

Abstract

It is well known that in hamiltonian systems the existence of transversal homoclinic orbit to a hyperbolic periodic solution leads to complicated behavior of phase trajectories and nonintegrability1,2. However, if we take an unstable equilibrium instead of a periodic solution, the situation is much more complicated. Devaney3 gave an example (the Neumann problem) of integrable system with transversal homoclinic orbit to the saddle equilibrium point.

Keywords

Manifold Torque Sine Librium 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. V. Bolotin, Liouville nonitegrability condition for a hamiltonian system, Vest. Mosc.Univ. Ser. Mat. Mech., 3:58 (1986).MathSciNetGoogle Scholar
  2. 2.
    S. L. Ziglin, Separatrices splitting and first integrals nonexistence in Hamiltonian system with two degrees of freedom, Izv. Akad. Nauk SSSR Ser. Mat., 51:1088 (1987).Google Scholar
  3. 3.
    R. L. Devaney, Homoclinic orbits in Hamiltonian systems, American J. Math., 100:631 (1978).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    R. L. Devaney, Transversal homoclinic orbits in an integrable system, J. Diff. Eqs., 21:431 (1976).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    L. M. Lerman and Ya. L. Umanskii, About separatices loop existence in four dimensional systems close to integrable hamiltonian ones, Prikh. Math. Mech., 47:395 (1983).MathSciNetGoogle Scholar
  6. 6.
    S. V. Bolotin, Double-asymptotic solutions and nonitegrability conditions of a hamiltonian system, Vest.Mosc.Univ.Ser.Mat.Mech., 1:55 (1990).MathSciNetGoogle Scholar
  7. 7.
    D. W. Turaev and L. P. Shilnikov, On Hamiltonian systems with homoclinic saddle curves, Dokl.Acad.Nauk SSSR, 304:811 (1989).MathSciNetGoogle Scholar
  8. 8.
    G. Benttin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems, Meccanica, March:9 (1980).Google Scholar
  9. 9.
    V. V. Beletskii, “The motion of an artificial satellite around the center of mass”, Nauka, Moscow, (1965).Google Scholar
  10. 10.
    A. P. Markeev, Resonance effects and stability of a satellite stationary rotations, Kosm. Issled., 5:365 (1967).Google Scholar
  11. 11.
    A. P. Markeev, Rotations of a satellite on elliptic orbit, Kosm. Issled., 5:530 (1967).Google Scholar
  12. 12.
    A. P. Markeev, On periodic motions of a satellite, Kosm. Issled., 23:323 (1985).Google Scholar
  13. 13.
    A. P. Markeev, Asymptotic trajectories and stability of periodic motions of an autonomous hamiltonian system, Prikh. Mat. Mech., 52:363 (1988).MathSciNetGoogle Scholar
  14. 14.
    A. G. Sokolsky, Stability of regular precessions, Kosm. Issled., 28:698 (1980).Google Scholar
  15. 15.
    Yu. W. Barkin, Skewed regular motions of a satellite and some small effects in the motion of the Moon and Phobos, Kosm. Issled., 23:26 (1985).Google Scholar
  16. 16.
    M. W. Demin, Plane periodical motion of the satellite around the mass center in the vincinity of the trigonal libration point, Kossm. Issled., 27:347 (1989).Google Scholar
  17. 17.
    J. M. A. Danby, Two notes on the Copenhagen problem, Celest. Mech., 33:251 (1984).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    J. M. A. Danby, The evolution of periodic orbits close to heteroclinic points, Celest. Mech., 33:26 (1984).Google Scholar
  19. 19.
    S. A. Dovbysh, Numerical investigations of the transversal separatrices crossing and the Kolmogorov stability, in: “Numerical analyzis, mathematical modeling in mechanics”, B. E. Poberdi, ed., Moscow State Univ., Moscow (1988).Google Scholar
  20. 20.
    K. Goździewski and A. J. Maciejewski, System for normalization of a Hamiltonian function, Celest. Mech., 49:1 (1990).MATHCrossRefGoogle Scholar
  21. 21.
    A. J. Maciejewski and K. Goździewski, Normalization algorithms of Hamiltonian near an equilibrium point, Astoph. Space Sci., 179:1 (1990).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Andrzej J. Maciejewski
    • 1
  • Krzysztof Goździewski
    • 1
  1. 1.Institute of AstronomyNicolaus Copernicus UniversityToruń, ChopinaPoland

Personalised recommendations