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On motions of a conservative system on invariant manifolds

Valentin Irtegov, Tatiana Titorenko
In [Irtegov and Burlakova, 2017], the algorithms for the qualitative analysis of conservative systems have been presented. These are based on the Routh-Lyapunov method [Lyapunov, 1954] and some its modifications [Irtegov and Titorenko, 2009] as well as computer algebra methods [Cox, Little, and O’Shea, 1997]. In the paper the application of the algorithms is demonstrated by analysing a conservative system, the study of which is also of interest. We conduct qualitative analysis for the differential equations describing the rotational motion of a rigid body with a fixed point in two constant force fields. Similar problems arise, e.g., in space dynamics [Sarychev and Gutnik, 2015], quantum mechanics [Adler, Marikhin, and Shabat, 2012], [Smirnov, 2008]. In the phase space of the problem, we isolate the invariant manifolds of maximal dimension and study the equations of motion on them.
For these equations, solutions (and their families) corresponding in the original phase space of the problem to permanent rotations and pendulum-like oscillations of the body as well as the invariant manifolds of 2nd level, which these solutions belong to, have been found and their Lyapunov’s stability has been investigated. The possibility of stabilization for the motions of conservative systems, whose stability conditions have the form of some constraints on the constants of first integrals, is discussed.
CYBERNETICS AND PHYSICS, Vol. 7, No. 3. 2018, 130–143. https://doi.org/10.35470/2226-4116-2018-7-3-130-143
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