Abstract
The problem of studying the stability of equilibrium solution of the second order non-autonomous Hamiltonian system, containing a small parameter, is considered. The main steps in solving this problem and application of the computer algebra systems for doing necessary calculations are discussed. As an example, we analyze stability of some equilibrium solutions in the elliptic restricted (2n + 1)-body problem. The problem is solved in a strict nonlinear formulation. All calculations are done with the computer algebra system Mathematica.
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References
Markeev, A.P.: The stability of hamiltonian systems (in Russian). In: Matrosov, V.M., Rumyantsev, V.V., Karapetyan, A.V. (eds.) Nonlinear meczhanics, Moscow, Fizmatlit, pp. 114–130 (2001)
Grebenikov, E.A., Kozak-Skovorodkin, D., Jakubiak, M.: The methods of computer algebra in the many-body problem (in Russian). Moscow, Ed. RUDN (2002)
Kolmogorov, A.N.: On the conservation of quasi-periodic motions for a small change in the Hamiltonian function (in Russian). Dokl. Akad. Nauk USSR 98(4), 527–530 (1954)
Arnold, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics (in Russian). Uspekhi Math. Nauk 18(6), 91–192 (1963)
Moser, J.: Lectures on the Hamiltonian systems. Moscow, Mir (1973)
Grebenikov, E.A., Gadomski, L., Prokopenya, A.N.: Studying the stability of equilibrium solutions in the planar circular restricted four-body problem. Nonlinear oscillations 10(1), 66–82 (2007)
Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press (1999)
Grebenikov, E.A., Prokopenya, A.N.: On existence of a new class of exact solutions in the planar Newtonian many-body problem (in Russian). In: The questions of modelling and analysis in the problems of making decision. Dorodnicyn Computing Center of RAS, Moscow, pp. 39–56 (2004)
Grebenikov, E.A., Prokopenya, A.N.: Determination of the boundaries between the domains of stability and instability for the Hill equation. Nonlinear Oscillations 6(1), 42–51 (2003)
Birkhoff, G.D.: Dynamical systems. Moscow (in Russian), GITTL (1941)
Prokopenya, A.N.: Normalization of the non-autonomous linear Hamiltonian system with a small parameter (in Russian). Mathematical Modelling 17(6), 33–42 (2005)
Arnold, V.I.: On stability of equilibrium positions of the Hamiltonian system of ordinary differential equations in general elliptic case (in Russian). Dokl. Akad. Nauk USSR 137(2), 255–257 (1961)
Markeev, A.P.: Libration points in celestial mechanics and cosmodynamics (in Russian). Moscow, Nauka (1978)
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Grebenikov, E.A., Ikhsanov, E.V., Prokopenya, A.N. (2007). Studying the Stability of the Second Order Non-autonomous Hamiltonian System. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2007. Lecture Notes in Computer Science, vol 4770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75187-8_15
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DOI: https://doi.org/10.1007/978-3-540-75187-8_15
Publisher Name: Springer, Berlin, Heidelberg
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