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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Beletskii, V.V.: The motion of an artificial satellite about its center of mass in a gravitational field. Moscow Univ. Press (1975) (in Russian)
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)
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)
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)
Markeev, A.P.: The libration points in celestial mechanics and cosmic dynamics, Nauka, Moscow (1978) (in Russian)
Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press, Cambridge (1999)
Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients. John Wiley, New York (1975)
Lindh, K.G., Likins, P.W.: Infinite determinant methods for stability analysis of periodic-coefficient differential equations. AIAA J. 8, 680–686 (1970)
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)
Prokopenya, A.N.: Determination of the stability boundaries for the hamiltonian systems with periodic coefficients. Math. Modelling and Analysis 10(2), 191–204 (2005)
Liapunov, A.M.: General problem about the stability of motion, Gostekhizdat, Moscow (1950) (in Russian)
Merkin, D.R.: Introduction to the Theory of Stability. Springer, Berlin (1997)
Grimshaw, R.: Nonlinear Ordinary Differential Equations. CRC Press, Boca Raton (2000)
Cesari, L.: Asymptotic behaviour and stability problems in ordinary differential equations, 2nd edn. Academic Press, New York (1964)
Landau, L.D., Lifshits, E.M.: Theoretical Physics, 4th edn. Mechanics, vol. 1. Nauka, Moscow (1988) (in Russian)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cattani, C., Grebenikov, E.A., Prokopenya, A.N. (2005). Symbolic Calculations in Studying the Stability of Dynamically Symmetric Satellite Motion. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_8
Download citation
DOI: https://doi.org/10.1007/11555964_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28966-1
Online ISBN: 978-3-540-32070-8
eBook Packages: Computer ScienceComputer Science (R0)