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Mathematics and Mechanics

Russian Academy of Sciences
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in January 1936
(Translated from 1958)
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ISSN 0021-8928
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IssuesArchive of Issues2009-3pp.249-258

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B.S. Bardin and A.M. Chekin, "Non-linear oscillations of a Hamiltonian system in the case of 3:1 resonance," J. Appl. Math. Mech. 73 (3), 249-258 (2009)
Year 2009 Volume 73 Issue 3 Pages 249-258
Title Non-linear oscillations of a Hamiltonian system in the case of 3:1 resonance
Author(s) B.S. Bardin (Moscow, Russia, bsbardin@yandex.ru)
A.M. Chekin (Moscow, Russia)
Abstract The motion of an autonomous Hamiltonian system with two degrees of freedom near its equilibrium position is considered. It is assumed that, in a certain region of the equilibrium position, the Hamiltonian is an analytic and sign-definite function, while the frequencies of linear oscillations satisfy a 3:1 ratio. A detailed analysis of the truncated system, corresponding to the normalized Hamiltonian is given, in which terms of higher than the fourth order are dropped. It is shown that the truncated system can be integrated in terms of Jacobi elliptic functions, and its solutions describe either periodic motions or motions that are asymptotic to periodic motions, or conventionally periodic motions. It is established, using the KAM-theory methods, that the majority of conventionally periodic motions are also preserved in the complete system. Moreover, in a fairly small neighbourhood of the equilibrium position, the trajectories of the complete system, which are not conventionally periodic, form a set of exponentially small measure. The results of the investigation are used in the problem of the motion of a dynamically symmetrical satellite in the region of its cylindrical precession.
Received 25 September 2008
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