Skip to main content
SpringerLink
Log in

Non-Integrability of the Problem of a Rigid Satellite in Gravitational and Magnetic Fields

  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

In this paper we analyse the integrability of a dynamical system describing the rotational motion of a rigid satellite under the influence of gravitational and magnetic fields. In our investigations we apply an extension of the Ziglin theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric satellite the system does not admit an additional real meromorphic first integral except for one case when the value of the induced magnetic moment along the symmetry axis is related to the principal moments of inertia in a special way.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V. I.: 1978, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Springer, New York.

    Google Scholar 

  2. Audin, M.: 2001, 'Les systèmes hamiltoniens et leur intégrabilité', Cours Spécialisés. SMF et EDP Sciences.

  3. Audin, M.: 2002, La réduction symplectique appliquée á la non-intégrabilité du problème du satellite, 2002.

  4. Baider, A., Churchill, R. C., Rod, D. L. and Singer, M. F.: 1996, 'On the infinitesimal geometry of integrable systems'. In: Mechanics Day (Waterloo, ON, 1992), Vol. 7 of Fields Inst. Commun., American Mathematical Society, Providence, RI, pp. 5-56.

    Google Scholar 

  5. Beletskii, V. V.: 1965, Motion of a Satellite about its Mass Center, Nauka, Moscow (in Russian).

    Google Scholar 

  6. Beletskii, V. V.: 1975, Motion of a Satellite about its Mass Center in the Gravitational Field, Moscow University Press, Moscow (in Russian).

    Google Scholar 

  7. Beletskii, V. V. and Hentov, A. A.: 1985, Rotational Motion of a Magnetized Satellite, Moscow University Press, Moscow (in Russian).

    Google Scholar 

  8. Beukers, F.: 1992, 'Differential Galois theory'. In: From Number Theory to Physics (Les Houches, 1989), Springer, Berlin, pp. 413-439.

    Google Scholar 

  9. Bogoyavlenskii, O. I.: 1985, 'Integrable cases of rigid-body dynamics and integrable systems on spheres S n', Izv. Akad. Nauk SSSR Ser. Mat. 49(5), 899-915.

    Google Scholar 

  10. Boucher, D.: 2000, 'Sur les équations différentielles linéaires paramétrées, une application aux systèmes hamiltoniens', PhD Thesis, Universitè de Limoges, France.

    Google Scholar 

  11. de Brun, F.: 1983, 'Rotation kring fix punkt', Öfvers. Kongl. Svenska Vetenskaps-Akad. Föhandl. 7, 455-468.

    Google Scholar 

  12. Duboshin, G. N.: 1968, Celestial Mechanics: Fundamental Problems and Methods, 2nd, Revised and Enlarged edn, Nauka, Moscow (in Russian).

    Google Scholar 

  13. Duval, A. and Loday-Richaud, M.: 1989, 'A propos de l'algoritme de Kovačic', Technical Report, Université de Paris-Sud,Mathématiques, Orsay, France.

    Google Scholar 

  14. Duval, A. and Loday-Richaud, M.: 1992, Kovačič's algorithm and its application to some families of special functions', Appl. Algebra Engrg. Comm. Comput. 3(3), 211-246.

    Google Scholar 

  15. Ince, E. L.: 1944, Ordinary Differential Equations, Dover, New York.

    Google Scholar 

  16. Kaplansky, I.: 1976, An Introduction to Differential Algebra, 2nd edn, Hermann, Paris; Actualités Scientifiques et Industrielles, No. 1251, Publications de l'Institut de Mathématique de l'Université de Nancago, No. V.

    Google Scholar 

  17. Kovacic, J. J.: 1986, 'An algorithm for solving second order linear homogeneous differential equations', J. Symbolic Comput. 2(1), 3-43.

    Google Scholar 

  18. Kozlov, V. V.: 1996, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, Berlin.

    Google Scholar 

  19. Maciejewski, A. J.: 1997, 'A simple model of the rotational motion of a rigid satellite around an oblate planet', Acta Astronomica 47, 387-398.

    Google Scholar 

  20. Maciejewski, A. J.: 2001, 'Non-integrability in gravitational and cosmological models, Introduction to Ziglin theory and its differential Galois extension', In: A. J. Maciejewski and B. Steves (eds), The Restless Universe: Applications of Gravitational N-Body Dynamics to Planetary, Stellar and Galatic Systems, pp. 361-385.

  21. Maciejewski, A. J.: 2001, 'Non-integrability of a certain problem of rotational motion of a rigid satellite', In: H. Prętka-Ziomek, E. Wnuk, P. K. Seidelmann and D. Richardson (eds), Dynamics of Natural and Artificial Celestial Bodies, Kluwer Academic Publisher, Dordrecht, pp. 187-192.

    Google Scholar 

  22. Magid, A. R.: 1994, Lectures on Differential Galois Theory, Vol. 7 of University Lecture Series, American Mathematical Society, Providence, RI.

    Google Scholar 

  23. Marsden, J. E. and Ratiu, T. S.: 1999, Introduction to Mechanics and Symmetry, Number 17 in Texts in Applied Mathematics, 2nd edn, Springer, New York, Berlin, Heidelberg.

    Google Scholar 

  24. Morales-Ruiz, J. J.: 2000, 'Kovalevskaya, Liapounov, Painlevé, Ziglin and the differential Galois theory', Regul. Chaotic Dyn. 5(3), 251-272.

    Google Scholar 

  25. Morales-Ruiz, J. J.: 1999, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Birkhäuser, Basel.

    Google Scholar 

  26. Morales-Ruiz, J. J. and Ramis, J. P.: 2001, 'Galoisian obstructions to integrability of Hamiltonian systems I, II', Meth. Appl. Anal. 8(1), 33-95, 97-111.

    Google Scholar 

  27. Poincaré, H.: 1890, 'Sur le problème des trois corps et les équations de la dynamique', Acta Math. 13, 1-270.

    Google Scholar 

  28. Singer, M. F.: 1990, 'An outline of differential Galois theory', In: Computer Algebra and Differential Equations, Academic Press, London, pp. 3-57.

    Google Scholar 

  29. Singer, M. F. and Ulmer, F.: 1993, 'Liouvillian and algebraic solutions of second and third order linear differential equations', J. Symbolic Comput. 16(1), 37-73.

    Google Scholar 

  30. Singer, M. F. and Ulmer, F.: 1995, 'Necessary conditions for Liouvillian solutions of (third order) linear differential equations', Appl. Algebra Eng. Comm. Comput. 6(1), 1-22.

    Google Scholar 

  31. Tsygvintsev, A.: 2000, 'La non-intégrabilité méromorphe du problème plan des trois corps', CR Acad. Sci. Paris Sér. I Math. 331(3), 241-244.

    Google Scholar 

  32. Tsygvintsev, A.: 2001, 'The meromorphic non-integrability of the three-body problem', J. Reine Angew. Math. 537, 127-149.

    Google Scholar 

  33. Tsygvintsev, A.: 2001, 'Sur l'absence d'une intégrale première méromorphe supplémentaire dans le problème plan des trois corps', CR Acad. Sci. Paris S´er. I Math. 333(2), 125-128.

    Google Scholar 

  34. Ulmer, F. and Weil, J.-A.: 1996, 'Note on Kovacic's algorithm', J. Symbolic Comput. 22(2), 179-200.

    Google Scholar 

  35. van Hoeij, M., Ragot, J.-F., Ulmer, F. and Weil, J.-A.: 1998, 'Liouvillian solutions of linear differential equations of order three and higher', J. Symbolic Comput. 28, 589-609.

    Google Scholar 

  36. Whittaker, E. T. and Watson, G. N.: 1965, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies with an Introduction to the Problem of Three Bodies, 4th edn, Cambridge University Press, London.

    Google Scholar 

  37. Ziglin, S. L.: 1982, 'Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. I', Funct. Anal. Appl. 16, 181-189.

    Google Scholar 

  38. Ziglin, S. L.: 1983, 'Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. II', Funct. Anal. Appl. 17, 6-17.

    Google Scholar 

  39. Ziglin, S. L.: 1997, 'On the absence of a real-analytic first integral in some problems of dynamics', Funktsional. Anal. i Prilozhen. 31(1), 3-11, 95.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Astronomy, University of Zielona Góra, Podgórna 50, PL-65–246, Zielona Góra, Poland, e-mail

    Andrzej J. Maciejewski

  2. INRIA Projet CAFÉ, 2004, Routes des Lucioles, B.P. 93, 06902 Sophia, Antipolis, Cedex, France

    Maria Przybylska

  3. Toruń Centre for Astronomy, Nicholaus Copernicus University, Gagarina 11, PL-87–100, Toruń, Poland, e-mail

    Maria Przybylska

Authors
  1. Andrzej J. Maciejewski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Przybylska
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maciejewski, A.J., Przybylska, M. Non-Integrability of the Problem of a Rigid Satellite in Gravitational and Magnetic Fields. Celestial Mechanics and Dynamical Astronomy 87, 317–351 (2003). https://doi.org/10.1023/B:CELE.0000006716.58713.ae

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:CELE.0000006716.58713.ae

Search

Navigation