 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| A. P. Markeev, "Stability of equilibrium states of hamiltonian systems: a method of investigation," Mech. Solids. 39 (6), 1-8 (2004) |
| Year |
2004 |
Volume |
39 |
Number |
6 |
Pages |
1-8 |
| Title |
Stability of equilibrium states of hamiltonian systems: a method of investigation |
| Author(s) |
A. P. Markeev (Moscow) |
| Abstract |
This paper deals with a time-periodic Hamiltonian system
with a single degree of freedom. It is assumed that the system has a state of equilibrium and the Hamiltonian function is analytic in its neighborhood. An efficient algorithm is proposed for obtaining the stability conditions for this equilibrium. This algorithm is based on the procedure of constructing and analyzing a simplectic mapping of the equilibrium neighborhood onto itself. All possible resonances up to the fourth order are considered, as well as the case of the absence of such resonances. Sufficient conditions for stability and instability are expressed in terms of the coefficients of the generating function of the mapping. As an application, the stability problem is solved in the special case of a satellite rotating about the center of mass in an elliptic orbit. |
| References |
| 1. | A. P. Markeev, "Stability of Hamiltonian systems," in Nonlinear Mechanics [in Russian], Fizmatgiz, Moscow, 2001, pp. 114-130. |
| 2. | A. P. Markeev, Libration Points in Celestial Mechanics and Cosmodynamics [in Russian], Nauka, Moscow, 1978. |
| 3. | T. Levi-Civita, "Sopra alcuni criteri di instabilita," Ann. Mat. Pura e Appl., Ser. 3, Vol. 5, pp. 221-307, 1901. |
| 4. | C. L. Siegel, "Vorlesungen \"uber Himmelsmechanik, Springer-Verlag, Berlin, 1956. |
| 5. | A. A. Markeev, "The method of pointwise mappings in the stability problem of two-segment trajectories of the Birkhoff billiards," Dynamical Systems in Classical Mechanics. Adv. Math. Sci. Amer. Math. Soc., Ser. 2,
Vol. 168, pp. 211-226, 1995. |
| 6. | A. P. Markeev, "On area-preserving mappings and their applications in dynamics of systems with collisions," Izv. AN. MTT [Mechanics of Solids], No. 2, pp. 37-54, 1996. |
| 7. | A. P. Markeev, Theoretical Mechanics [in Russian],
Regular and Chaotic Dynamics, Moscow, Izhevsk, 2001. |
| 8. | I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka, Moscow, 1966. |
| 9. | V. V. Beletskii, Motion of a Satellite Relative to the Center of Mass [in Russian], Nauka, Moscow, 1965. |
| 10. | A. A. Khentov, "On the stability of a rotation of a satellite about its center of mass in the first approximation," Kosmicheskie Issledovaniya [Cosmic Research], Vol. 6, No. 5, pp. 793-795, 1968. |
| 11. | A. P. Markeev and B. S. Bardin,
"On a plane rotation of a satellite in an elliptic orbit,"
Kosmicheskie Issledovaniya [Cosmic Research], Vol. 32, No. 6, pp. 43-49, 1994. |
| 12. | G. E. O. Giacaglia, Perturbation methods
in Nonlinear Systems, Springer-Verlag, Berlin, 1972. |
| 13. | J. K. Moser, Lectures on Hamiltonian Systems, Amer. Math. Soc., Provicence, R. I., 1968. |
|
| Received |
10 December 2003 |
| << Previous article | Volume 39, Issue 6 / 2004 | Next article >> |
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