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.
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The work was supported by a grant from the Russian Science Foundation (project no. 19-11-00116) at the Moscow Aviation Institute (National Research University).
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In loving memory of L.D. Akulenko
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Bardin, B.S., Maksimov, B.A. 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, 2894–2907 (2023). https://doi.org/10.3103/S0025654423080046
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DOI: https://doi.org/10.3103/S0025654423080046