 | | 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
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| B.S. Bardin and B.A. Maksimov, "On the Orbital Stability of Pendulum Periodic Motions of a Heavy Rigid Body with a Fixed Point in the Case of Principal Inertia Moments Ratio 1 : 4 : 1," Mech. Solids. 58 (8), 2894-2907 (2023) |
| Year |
2023 |
Volume |
58 |
Number |
8 |
Pages |
2894-2907 |
| DOI |
10.3103/S0025654423080046 |
| Title |
On the Orbital Stability of Pendulum Periodic Motions of a Heavy Rigid Body with a Fixed Point in the Case of Principal Inertia Moments Ratio 1 : 4 : 1 |
| Author(s) |
B.S. Bardin (Moscow Aviation Institute, Moscow, 125993 Russia, bsbardin@yandex.ru)
B.A. Maksimov (Moscow Aviation Institute, Moscow, 125993 Russia, badmamaksimov1@gmail.com) |
| Abstract |
The motion of a heavy rigid body with a fixed point in a uniform gravitational field is considered. It is assumed that the principal moments of inertia of the body for the fixed point satisfy the condition of Goryachev–Chaplygin; i.e., they are in the ratio 1 : 4 : 1. In contrast to the integrable case of Goryachev–Chaplygin, no additional restrictions are imposed on the position of the center of mass
of the body. The problem of orbital stability of pendulum periodic motions of the body is investigated.
In the neighborhood of periodic motions, local variables are introduced and equations of perturbed
motion are obtained. On the basis of a linear analysis of stability, the orbital instability of pendulum
rotations for all values of the parameters has been proven. It has been established that, depending on
the values of the parameters, pendulum oscillations can be both orbitally unstable and orbitally stable
in a linear approximation. For pendulum oscillations that are stable in the linear approximation, based
on the methods of KAM theory, a nonlinear analysis is performed and rigorous conclusions about the orbital stability are obtained. |
| Keywords |
pendulum periodic motions, orbital stability, Goryachev–Chaplygin case, local variables, Hamiltonian systems |
| Received |
10 June 2023 | Revised |
07 July 2023 | Accepted |
20 July 2023 |
| Link to Fulltext |
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