Advertisement

Symbolic Calculations in Studying the Stability of Dynamically Symmetric Satellite Motion

  • Carlo Cattani
  • Evgenii A. Grebenikov
  • Alexander N. Prokopenya
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)

Abstract

The stability of cylindrical precession of the dynamically symmetric satellite in the Newtonian gravitational field is studied. We consider the case when a center of mass of the satellite moves in an elliptic orbit, while the satellite rotates uniformly about the axis of its dynamical symmetry that is perpendicular to the orbit plane. In the case of the resonance 3:2 (Mercury type resonance) we have found the domains of instability of cylindrical precession of the satellite in the Liapunov sense and domains of its linear stability in the parameter space. Using the infinite determinant method we have calculated analytically the boundaries of the domains of instability as power series in the eccentricity of the orbit. All the calculations have been done with the computer algebra system Mathematica.

Keywords

Stability Boundary Orbit Plane Computer Algebra System Elliptic Orbit Characteristic Exponent 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gerdt, V.P., Tarasov, O.V., Shirkov, D.V.: Analytic calculations on digital computers for applications in physics and mathematics. Usp. Fiz. Nauk 130(1), 113–147 (1980) (in Russian)MathSciNetGoogle Scholar
  2. 2.
    Beletskii, V.V.: The motion of an artificial satellite about its center of mass in a gravitational field. Moscow Univ. Press (1975) (in Russian)Google Scholar
  3. 3.
    Markeev, A.P., Chekhovskaya, T.N.: On the stability of cylindrical precession of a satellite on the elliptic orbit. Prikl. Math. Mech. 40, 1040–1047 (1976) (in Russian) Google Scholar
  4. 4.
    Churkina, T.E.: On stability of a satellite motion in the elliptic orbit in the case of cylindrical precession. Mathematical Modelling 16(7), 3–5 (2004) (in Russian)zbMATHGoogle Scholar
  5. 5.
    Markeev, A.P.: The stability of hamiltonian systems. In: Matrosov, V.M., Rumyantsev, V.V., Karapetyan, A.V. (eds.) Nonlinear mechanics, Fizmatlit, Moscow, pp. 114–130 (2001) (in Russian)Google Scholar
  6. 6.
    Markeev, A.P.: The libration points in celestial mechanics and cosmic dynamics, Nauka, Moscow (1978) (in Russian)Google Scholar
  7. 7.
    Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  8. 8.
    Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients. John Wiley, New York (1975)zbMATHGoogle Scholar
  9. 9.
    Lindh, K.G., Likins, P.W.: Infinite determinant methods for stability analysis of periodic-coefficient differential equations. AIAA J. 8, 680–686 (1970)zbMATHCrossRefGoogle Scholar
  10. 10.
    Prokopenya, A.N.: Studying stability of the equilibrium solutions in the restricted many-body problems. In: Mitic, P., Ramsden, P., Carne, J. (eds.) Challenging the Boundaries of Symbolic Computation, Proc. 5th Int. Mathematica Symposium, London, Great Britain, pp. 105–112. Imperial College Press, London (2003)CrossRefGoogle Scholar
  11. 11.
    Prokopenya, A.N.: Determination of the stability boundaries for the hamiltonian systems with periodic coefficients. Math. Modelling and Analysis 10(2), 191–204 (2005)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Liapunov, A.M.: General problem about the stability of motion, Gostekhizdat, Moscow (1950) (in Russian)Google Scholar
  13. 13.
    Merkin, D.R.: Introduction to the Theory of Stability. Springer, Berlin (1997)Google Scholar
  14. 14.
    Grimshaw, R.: Nonlinear Ordinary Differential Equations. CRC Press, Boca Raton (2000)Google Scholar
  15. 15.
    Cesari, L.: Asymptotic behaviour and stability problems in ordinary differential equations, 2nd edn. Academic Press, New York (1964)Google Scholar
  16. 16.
    Landau, L.D., Lifshits, E.M.: Theoretical Physics, 4th edn. Mechanics, vol. 1. Nauka, Moscow (1988) (in Russian) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Carlo Cattani
    • 1
  • Evgenii A. Grebenikov
    • 2
  • Alexander N. Prokopenya
    • 3
  1. 1.DiFarmaUniversity of SalernoFisciano (SA)Italy
  2. 2.Computing Center of RASMoscowRussia
  3. 3.Brest State Technical UniversityBrestBelarus

Personalised recommendations