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 Regul. Chaotic Dyn., 2017, Volume 22, Issue 7, Pages 792–807 (Mi rcd291)

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

Olga V. Kholostovaab, Alexey I. Safonovc

a Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993 Russia
b Moscow Institute of Physics and Technology (State University), Institutskiy per. 9, Dolgoprudny, 141701 Russia
c JSC “NPF “Infosistem-35”, 3rd Mytischinskaya st. 16, bld. 37, Moscow, 129626 Russia

Abstract: We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.

Keywords: Hamiltonian system, resonance, stability, KAM theory, cylindrical precession of a satellite

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 3.3858.2017/4.6 This work was carried out within the framework of the state assignment (project ¹ 3.3858.2017/4.6).

DOI: https://doi.org/10.1134/S1560354717070036

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MSC: 70H08, 70H12, 70H14, 70H15, 70M20
Accepted:08.11.2017
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Citation: Olga V. Kholostova, Alexey I. Safonov, “A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance”, Regul. Chaotic Dyn., 22:7 (2017), 792–807

Citation in format AMSBIB
\Bibitem{KhoSaf17} \by Olga V. Kholostova, Alexey I. Safonov \paper A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance \jour Regul. Chaotic Dyn. \yr 2017 \vol 22 \issue 7 \pages 792--807 \mathnet{http://mi.mathnet.ru/rcd291} \crossref{https://doi.org/10.1134/S1560354717070036} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000425980500003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85042468014} 

• http://mi.mathnet.ru/eng/rcd291
• http://mi.mathnet.ru/eng/rcd/v22/i7/p792

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. S. Bardin, “On the stability of a periodic Hamiltonian system with one degree of freedom in a transcendental case”, Dokl. Math., 97:2 (2018), 161–163
2. A. P. Markeev, “On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession”, Nelineinaya dinam., 14:4 (2018), 503–518