Abstract
We construct the method of approximate solution for the differential equation of oscillations of a satellite on elliptic orbits subject to the gravity torque and the light-pressure torque. Parabolic orbits are included as a limiting case. The metric of a weighted Sobolev space is used as a measure of the vicinity. This allows us to construct a uniform approximation of a solution with respect to the eccentricity of the orbit.
To prove such an approximation, we use the Leray-Schauder degree theory and the Krasnosel’skij theorem of Galerkin approximations for compact vector fields adapted to the problem under consideration. To establish the uniform estimate of the convergence of approximate solutions to the solution, we also use a modification of an appropriate Krasnosel’skij theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. V. Beletskii, Motion of a Satellite around the Center of Mass in a Gravitational Field [in Russian], MGU, Moscow (1975).
G. N. Duboshin, Celestial Mechanics. Basic Problems and Methods [in Russian], Nauka, Moscow (1968).
G. M. Fikhtengol’ts, Course of Differential and Integral Calculus. Vol. 1 [in Russian], Lan’, Saint-Petersburg (1997).
A. A. Karymov, “Determination of forces and moments due to light pressure acting on a body in motion in cosmic space,” PMM, J. Appl. Math. Mech., 26, 1310–1324 (1963).
M. A. Krasnosel’skij, Topological Methods in the Theory of Nonlinear Integral Equations, International Series of Monographs on Pure and Applied Mathematics, 45, Pergamon Press, Oxford (1964).
M. A. Krasnosel’skij, P. P. Zabrejko, E. I. Pustylnik, and P. E. Sobolevskij, Integral Operators in Spaces of Summable Functions, Monographs and Textbooks on Mechanics of Solids and Fluids, Noordhoff International Publishing, Leiden (1976).
J. Leray and J. Schauder, “Topologie et équations fonctionnelles,” Ann. Sci. École Norm. Sup. (4), 13, 45–78 (1934).
A. P. Markeev, Theoretical Mechanics [in Russian], Nauka, Moscow (1990).
V. G. Maz’ya, Sobolev Spaces, Springer-Verlag, Berlin (1985).
L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York (1974).
I. Ya. Novikov and S. B. Stechkin, “Basic wavelet theory,” Russ. Math. Surv., 53, No. 6, 1159–1231 (1998).
J. Schauder, “Zur Theorie stetiger Abbildungen in Funktionalräumen,” Math. Z, 26, No. 5, 47–65 (1927).
L. Schwarz, Analysis. Vol. 1 [in Russian], Mir, Moscow (1972).
V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1980).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.
Rights and permissions
About this article
Cite this article
Kosenko, I.I. Topological degree and approximation of solutions for nonregular problems of mechanics: Oscillations of satellites on elliptic orbits. J Math Sci 149, 1539–1566 (2008). https://doi.org/10.1007/s10958-008-0081-5
Issue Date:
DOI: https://doi.org/10.1007/s10958-008-0081-5