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Russian Academy of Sciences
in January 1936
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IssuesArchive of Issues2013-6pp.588-594

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V.N. Tkhai, "A mechanical system containing weakly coupled subsystems," J. Appl. Math. Mech. 77 (6), 588-594 (2013)
Year 2013 Volume 77 Issue 6 Pages 588-594
Title A mechanical system containing weakly coupled subsystems
Author(s) V.N. Tkhai (Moscow, Russia,
Abstract The concept of a mechanical system (model), containing coupled subsystems (MSCCS) is introduced. Examples of such a system are the Sunplanetssatellites system, a system of interacting moving objects, a system of translationally and rotationally moving celestial bodies, chains of coupled oscillators, Sommerfeld pendulums, spring systems, etc. The MSCCS subsystems and the entire system are analysed, and problems related to the investigation of the oscillations, bifurcation, stability, stabilization and resonance are stated. A solution of the oscillations problem is given for a class of MSCCS, described by reversible mechanical systems. It is proved that the autonomous MSCCS retains its family of symmetrical periodic motions (SPMs) under parametric perturbations, while in the periodic MSCCS a family of SPMs bifurcates by producing two families of SPMs. The two-body problem and the N-planet problem are investigated as applications. The generating properties of the two-body problem are established. For the N-planet problem it is proved that an (N+1)-parametric family of orbits exists, close to elliptic orbits of arbitrary eccentricity, the family being parametrized by energy integral constant, and a syzygy of planets occurs.
Received 25 October 2012
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