Section
|
Mechanics
|
Title
|
On the motions of a near-autonomous Hamiltonian system in the cases of two zero frequencies
|
Author(-s)
|
Kholostova O.V.a
|
Affiliations
|
Moscow Aviation Institutea
|
Abstract
|
We consider the motion of a near-autonomous, time-periodic two-degree-of- freedom Hamiltonian system in the vicinity of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small, and when it is zero, the system is autonomous. Suppose that in the autonomous case for a set of two other parameters, both frequencies of small linear oscillations of the system in the vicinity of the equilibrium are equal to zero, and the rank of the coefficient matrix of the linearized equations of perturbed motion is three, two, or one. We study the structure of the regions of stability and instability of the trivial equilibrium of the system in the vicinity of the resonant point of a three-dimensional parameter space, as well as the existence, number and stability (in a linear approximation) of periodic motions of the system that are analytic in integer or fractional powers of the small parameter. As an application, periodic motions of a dynamically symmetric satellite (solid) with respect to the center of mass are obtained in the vicinity of its stationary rotation (cylindrical precession) in a weakly elliptical orbit in the case of two zero frequencies under study, and their instability is proved.
|
Keywords
|
Hamiltonian system, normalization, zero frequencies, stability, dynamically symmetric satellite, cylindrical precession
|
UDC
|
531.36, 521.1
|
MSC
|
70H08, 70H12, 70H14, 70H15, 70M20
|
DOI
|
10.35634/vm200410
|
Received
|
1 July 2020
|
Language
|
Russian
|
Citation
|
Kholostova O.V. On the motions of a near-autonomous Hamiltonian system in the cases of two zero frequencies, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 672-695.
|
References
|
- Markeev A.P. Parametric resonance and nonlinear oscillations of heavy solid near its flat rotations, Izvestiya Rossiiskoi Akademii Nauk. Mekhanika Tverdogo Tela, 1995, issue 5, pp. 34-44 (in Russian).
- Kholostova O.V. Parametric resonance in the problem on satellite nonlinear oscillations in elliptic orbit, Kosmicheskie Issledovaniya, 1996, vol. 3, no. 3, pp. 312-316 (in Russian).
- Kholostova O.V. The periodic motions of a non-autonomous Hamiltonian system with two degrees of freedom at parametric resonance of the fundamental type J. Appl. Math. Mech., 2002, vol. 66, issue 4, pp. 529-538. https://doi.org/10.1016/S0021-8928(02)00071-0
- Markeev A.P. On a multiple resonance in linear Hamiltonian systems, Doklady Phyics, 2005, vol. 50, issue 5, pp. 278-282. https://doi.org/10.1134/1.1941506
- Markeev A.P. On one special case of parametric resonance in problems of celestial mechanics, Astron. Lett., 2005, vol. 31, no. 5, pp. 350-356. https://doi.org/10.1134/1.1922534
- Markeev A.P. Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass, Astron. Lett., 2005, vol. 31, no. 9, pp. 627-633. https://doi.org/10.1134/1.2039974
- Markeyev A.P., Multiple parametric resonance in Hamilton systems, J. Appl. Math. Mech., 2006, vol. 70, issue 2, pp. 176-194. https://doi.org/10.1016/j.jappmathmech.2006.06.001
- Markeev A.P. Lineinye gamil'tonovy sistemy i nekotorye zadachi ob ustoichivosti dvizheniya sputnika otnositel'no centra mass (Linear Hamiltonian systems and some problems on stability of motion of a satellite about its center of mass), Izhevsk: R&C Dynamics, Institute of Computer Science, 2009.
- Kholostova O.V. On periodic motions of a nonautonomous Hamiltonian system in one case of multiple parametric resonance, Nelineinaya Dinamika, 2017, vol. 13, issue 4, pp. 477-504 (in Russian). https://doi.org/10.20537/nd1704003
- Kholostova O.V. On periodic motions of a nearly autonomous Hamiltonian system in the occurrence of double parametric resonance, Mechanics of Solids, 2019, vol. 54, issue 2, pp. 211-233. https://doi.org/10.3103/S0025654419030154
- Kholostova O.V. On the motions of one near-autonomous Hamiltonian system at a 1:1:1 resonance, Regular and Chaotic Dynamics, 2019, vol. 24, issue 3, pp. 235-265. https://doi.org/10.1134/S1560354719030018
- Sokol'skii A.G. On stability of self-contained Hamiltonian system with two degrees of freedom in the case of zero frequencies, J. Appl. Math. Mech., 1981, vol. 45, issue 3, pp. 321-327. https://doi.org/10.1016/0021-8928(81)90060-5
- Markeev A.P. Tochki libratsii v nebesnoi mekhanike i kosmodinamike (Libration points in celestial mechanics and space dynamics), Moscow: Nauka, 1978.
- Malkin I.G. Nekotorye zadachi teorii nelineinykh kolebanii (Some problems in the theory of nonlinear oscillations), Moscow: URSS, 2004.
- Markeev A.P. On rotational motion of a dynamically symmetrical satellite in an elliptic orbit, Kosmicheskie Issledovaniya, 1967, vol. 5, issue 4, pp. 530-539 (in Russian).
- Beletskii V.V. Dvizhenie sputnika otnositel'no tsentra mass v gravitatsionnom pole (Satellite’s motion about the center of mass in a gravitational field), Moscow: Moscow State University, 1975.
|
Full text
|
|