 | | Journal of Applied
Mathematics and Mechanics
Russian Academy of Sciences | | Founded
in January 1936
(Translated from 1958)
Issued 6 times a year
ISSN 0021-8928
(print version) |
Archive of Issues
| Total articles in the database: | | 10522 |
| In Russian (ΟΜΜ): | | 9723
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| In English (J. Appl. Math. Mech.): | | 799 |
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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 |
| Link to Fulltext |
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