Journal of Applied Mathematics and Mechanics (about journal) Journal of Applied
Mathematics and Mechanics

Russian Academy of Sciences
 Founded
in January 1936
(Translated from 1958)
Issued 6 times a year
ISSN 0021-8928
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IssuesArchive of Issues2011-6pp.631-646

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A.P. Markeyev, "Non-linear oscillations of a 1:1 resonance Hamiltonian system," J. Appl. Math. Mech. 75 (6), 631-646 (2011)
Year 2011 Volume 75 Issue 6 Pages 631-646
Title Non-linear oscillations of a 1:1 resonance Hamiltonian system
Author(s) A.P. Markeyev (Moscow, Russia, markeev@ipmnet.ru)
Abstract The non-linear oscillations of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its stable equilibrium position are considered. It is assumed that the Hamilton function is sign-definite in the neighbourhood of the equilibrium position and that the values of the frequencies of its linear oscillations are equal or close to one another (1:1 resonance). The investigation is carried out using the example of the problem of the motion of a dynamically symmetrical rigid body (satellite) about its centre of mass in a circular orbit in a central Newtonian gravitational field. In this problem there is relative equilibrium of the rigid body in the orbital system of coordinates, for which its axis of dynamic symmetry is directed along the velocity vector of the centre of mass. Resonance occurs when the ratio of the polar and equatorial principal central moments of inertia is equal to 4/3 or is close to it. The problem of the existence, bifurcation and orbital stability of the periodic motions of a rigid body generated from its relative equilibrium is solved. Some aspects of the existence of quasiperiodic motions are also considered.
Received 25 January 2011
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