NDLI: Stability problem for pendulum-type motions of a rigid body in the Goryachev-Chaplygin case

Content Provider	Springer Nature Link
Author	Bardin, B. S.
Copyright Year	2007
Abstract	We consider the motion of a rigid body with a single fixed point in a homogeneous gravity field. The body mass geometry and the initial conditions for its motion correspond to the case of Goryachev—Chaplygin integrability. We study the orbital stability problem for periodic motions corresponding to vibrations and rotations of the rigid body rotating about the equatorial axis of the inertia ellipsoid.In [1], it was proved that these periodic motions are orbitally unstable in the linear approximation. It was also shown that, to solve the stability problem in the nonlinear setting, it does not suffice to analyze terms up to the fourth order in the expansion of the Hamiltonian function in the canonical variables.The present paper shows that in this problem one deals with a special case where standard methods for stability analysis based on the coefficients in the normal form of the Hamiltonian of the perturbed equations of motion do not apply. We use Chetaev’s theorem to prove the orbital instability of these periodic motions in the rigorous nonlinear statement of the problem. The proof uses the additional first integral of the Goryachev—Chaplygin problem in an essential way.
Starting Page	177
Ending Page	183
Page Count	7
File Format	PDF
ISSN	00256544
Journal	Mechanics of Solids
Volume Number	42
Issue Number	2
e-ISSN	19347936
Language	English
Publisher	Allerton Press, Inc.
Publisher Date	2007-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mechanics
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy Mechanics of Materials

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