First integrals and families of symmetric periodic motions of a reversible mechanical system

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Tkhai, V.N.
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Journal of Applied Mathematics and Mechanics | 2006年 / 70卷 / 06期
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A reversible mechanical system which allows of first integrals is studied. It is established that; for symmetric motions; the constants of the asymmetric integrals are equal to zero. The form of the integrals of a reversible linear periodic system corresponding to zero characteristic exponents and the structure of the corresponding Jordan Boxes are investigated. A theorem on the non-existence of an additional first integral and a theorem on the structural stabilities of having a symmetric periodic motion (SPM) are proved for a system with m symmetric and k asymmetric integrals. The dependence of the period of a SPM on the constants of the integrals is investigated. Results of the oscillations of a quasilinear system in degenerate cases are presented. Degeneracy and the principal resonance: bifurcation with the disappearance of the SPM and the birth of two asymmetric cycles; are investigated. A heavy rigid body with a single fixed point is studied as the application of the results obtained. The Euler-Poisson equations are used. In the general case; the energy integral and the geometric integral are symmetric while the angular momentum integral turns out to be asymmetric. In the special case; when the centre of gravity of the body lies in the principal plane of the ellipsoid of inertia; all three classical integrals become symmetric. It is ascertained here that any SPM of a body contains four zero characteristic exponents; of which two are simple and two form a Jordan Box. In typical situation; the remaining two characteristic exponents are not equal to zero. All of the above enables one to speak of an SPM belonging to a two-parameter family and the absence of an additional first integral. It is established that a body also executes a pendulum motion in the case when the centre of gravity is close to the principal plane of the ellipsoid of inertia. © 2007 Elsevier Ltd. All rights reserved;
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