We consider the motion of a heavy rigid body with a fixed point in a uniform gravitational field under the assumption that the principal moments of inertia satisfy the Goryachev–Chaplygin condition at the fixed point. We study the orbital stability problem for small pendulum oscillations of the body. We derive the equations of perturbed motion and reduce the problem to the study of the stability of the equilibrium position of a second order 2π-periodic Hamiltonian system. We find regions of parametric resonance and perform the nonlinear analysis of orbital stability outside these regions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. P. Markeev, “The stability of the plane motions of a rigid body in the Kovalevskaya case,” J. Appl. Math. Mech. 65, No. 1, 47–54 (2001).
A. Z. Bryum, “Orbital stability analysis using first integrals,” J. Appl. Math. Mech. 53, No. 6, 689-695 (1989).
A.Z. Bryum and A. Ya. Savchenko, “On the orbital stability of a periodic solution of the equations of motion of a Kovalevskaya gyroscope,” J. Appl. Math. Mech. 50, No. 6 , 748-753 (1986).
B. S. Bardin, “Stability problem for pendulum-type motions of a rigid body in the Goryachev–Chaplygin case,” Mech. Solids 42, No. 2, 177–183 (2007).
A. P. Markeev, “Pendulum-like motions of a solid in the Goryachev–Chaplygin case,” J. Appl. Math. Mech. 68, No. 2, 249-258 (2004).
B. S. Bardin, T. V. Rudenko, and A. A. Savin, “On the orbital stability of planar periodic motions of a rigid body in the Bobylev–Steklov case,” Regul. Chaotic Dyn. 17, No. 6, 533–546 (2012).
B. S. Bardin and A. A. Savin, “On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point,” Regul. Chaotic Dyn. 17, No. 3-4, 243–257 (2012).
B. S. Bardin and A. A. Savin, “ The stability of the plane periodic motions of a symmetrical rigid body with a fixed point,” J. Appl. Math. Mech. 77, No. 6, 578-587 (2013).
H. M. Yehia and E. G. El-Hadidy, “On the orbital stability of pendulum-like vibrations of a rigid body carrying a rotor,” Regul. Chaotic Dyn. 18, No. 5, 539–552 (2013).
B. S. Bardin, “On a method of introducing local coordinates in the problem of the orbital stability of planar periodic motions of a rigid body,” Rus. J. Nonlinear Dyn. 16, No. 4, 581–594 (2020).
A. P. Markeev, Linear Hamiltonian Systems and Some Problems of Stability of the Satellite Center of Mass [in Russian], Institute of Computer Research Press, Izhevsk (2009).
G. E. O. Giacaglia, Perturbation Methods in Non-Linear Systems, Springer, New York etc. (1972).
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York (1989).
A. P. Ivanov and A. G. Sokol’skij, “On the stability of a non-autonomous Hamiltonian system in parametric resonance of dominant mode,” J. Appl. Math. Mech., 44, No. 6, 687-691 (1980).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bardin, B.S., Maksimov, B.A. The Orbital Stability Analysis of Pendulum Oscillations of a Heavy Rigid Body with a Fixed Point Under the Goriachev–Chaplygin Condition. J Math Sci 275, 66–77 (2023). https://doi.org/10.1007/s10958-023-06660-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06660-2