 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
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| B. S. Bardin, "Stability problem for pendulum-type motions of a rigid body in the Goryachev-Chaplygin case," Mech. Solids. 42 (2), 177-183 (2007) |
| Year |
2007 |
Volume |
42 |
Number |
2 |
Pages |
177-183 |
| Title |
Stability problem for pendulum-type motions of a rigid body in the Goryachev-Chaplygin case |
| Author(s) |
B. S. Bardin (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993, Russia, bsbardin@yandex.ru) |
| 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. |
| References |
| 1. | A. P. Markeev,
"On the Identity Resonance in a Particular Case of the Problem
of Stability of Periodic Motions of a Rigid Body,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 32-37 (2003)
[Mech. Solids (Engl. Transl.)]. |
| 2. | D. N. Goryachev,
"On the Motion of a Heavy Rigid Body about a Fixed Point in the Case A=B=4C,"
Mat. Sb. Kruzhka Lyubitelei Mat. Nauk
21 (3), 431-438 (1900). |
| 3. | S. A. Chaplygin,
"A New Case of Rotation of a Heavy Rigid Body Supported at a Single Point,"
Tr. Otd. Fiz. Nauk O-va Lyubitelei Estestvoznan.
10 (2), 32-34 (1901). |
| 4. | G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova,
Classical Problems of Dynamics of Solids.
Development and Modern State
(Naukova Dumka, Kiev, 1978)
[in Russian]. |
| 5. | A. I. Doshkevich, Closede-Form Solutions of the
Euler-Poisson Equations (Naukova Dumka, Kiev, 1992) [in
Russian]. |
| 6. | V. V. Kozlov, Methods for Qualitative Analysis in
Dynamics of Solids, in Regular and Chaotic Dynamics
(NITs, Izhevsk, 2000) [in Russian]. |
| 7. | A. V. Borisov and I. S. Mamaev, Dynamics of Solids,
in Regular and Chaotic Dynamics (NITs, Izhevsk, 2001)
[in Russian]. |
| 8. | N. I. Akhiezer, Elements of Theory of Elliptic
Functions (Nauka, Moscow, 1970) [in Russian]. |
| 9. | I. G. Malkin, Theory of Stability of Motion (Nauka,
Moscow, 1966) [in Russian]. |
| 10. | A. P. Ivanov and A. G. Sokol'skii, "On the Stability of a
Nonautonomous Hamiltonian System Under a Parametric
Resonance of Essential Type," Prikl. Mat. Mekh. 44
(6), 963-970 (1980) [J. Appl. Math. Mech. (Engl.
Transl.)]. |
| 11. | G. A. Merman, "Asymptotic Solutions of the Canonical
System with One Degree of Freedom in the Case of Zero
Characteristic Exponents," Bull. Inst. Teoret. Astron. AN
SSSR 9 No. 6 (109), 394-424 (1964). |
| 12. | A. P. Markeev,
"Study of the Stability of Periodic Motions of an Autonomous
Hamiltonian System in a Critical Case,"
Prikl. Mat. Mekh.
64 (5), 833-847 (2000)
[J. Appl. Math. Mech. (Engl. Transl.)]. |
| 13. | B. S. Bardin, "On Asymptotic Solutions of Hamiltonian
System in the Case of First-Order Resonance," Prikl. Mat.
Mekh. 55 (4), 587-593 (1991) [J. Appl. Math. Mech.
(Engl. Transl.)]. |
|
| Received |
01 November 2005 |
| Link to Fulltext |
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