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Regular and Chaotic Dynamics

, Volume 22, Issue 7, pp 792–807 | Cite as

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

  • Olga V. KholostovaEmail author
  • Alexey I. Safonov
Article

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 

MSC2010 numbers

70H08 70H12 70H14 70H15 70M20 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Moscow Aviation Institute (National Research University)MoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University)DolgoprudnyRussia
  3. 3.JSC“NPF “Infosistem-35”MoscowRussia

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