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Russia Izhevsk
Year
2020
Volume
30
Issue
4
Pages
628-644
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Section Mechanics
Title Nonintegrability of the problem of a spherical top rolling on a vibrating plane
Author(-s) Kilin A.A.a, Pivovarova E.N.a
Affiliations Udmurt State Universitya
Abstract This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.
Keywords spherical top, vibrating plane, Lagrange case, separatrix splitting, Melnikov's integral, nonintegrability, chaos, period advance map
UDC 531.36
MSC 70E18, 37J30
DOI 10.35634/vm200407
Received 25 September 2020
Language Russian
Citation Kilin A.A., Pivovarova E.N. Nonintegrability of the problem of a spherical top rolling on a vibrating plane, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 628-644.
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