On the motions of a near-autonomous hamiltonian system in the cases of two zero frequencies
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 4, pp. 672-695
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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.
@article{VUU_2020_30_4_a9,
author = {O. V. Kholostova},
title = {On the motions of a near-autonomous hamiltonian system in the cases of two zero frequencies},
journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
pages = {672--695},
publisher = {mathdoc},
volume = {30},
number = {4},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VUU_2020_30_4_a9/}
}
TY - JOUR AU - O. V. Kholostova TI - On the motions of a near-autonomous hamiltonian system in the cases of two zero frequencies JO - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki PY - 2020 SP - 672 EP - 695 VL - 30 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/VUU_2020_30_4_a9/ LA - ru ID - VUU_2020_30_4_a9 ER -
%0 Journal Article %A O. V. Kholostova %T On the motions of a near-autonomous hamiltonian system in the cases of two zero frequencies %J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki %D 2020 %P 672-695 %V 30 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/VUU_2020_30_4_a9/ %G ru %F VUU_2020_30_4_a9
O. V. Kholostova. On the motions of a near-autonomous hamiltonian system in the cases of two zero frequencies. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 4, pp. 672-695. http://geodesic.mathdoc.fr/item/VUU_2020_30_4_a9/