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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.
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