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A Top on a Vibrating Base: New Integrable Problem of Nonholonomic Mechanics

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Abstract

A spherical rigid body rolling without sliding on a horizontal support is considered. The body is axially symmetric but unbalanced (tippe top). The support performs high-frequency oscillations with small amplitude. To implement the standard averaging procedure, we present equations of motion in quasi-coordinates in Hamiltonian form with additional terms of nonholonomicity [16] and introduce a new fast time variable. The averaged system is similar to the initial one with an additional term, known as vibrational potential [8, 9, 18]. This term depends on the single variable — the nutation angle \(\theta\), and according to the work of Chaplygin [5], the averaged system is integrable. Some examples exhibit the influence of vibrations on the dynamics.

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Funding

This work was supported by the Russian Foundation for Basic Research (project 18-29-10051).

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology , Inststitutskii per. 9, 141700, Dolgoprudnyi, Russia

    Alexander P. Ivanov

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  1. Alexey V. Borisov
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  2. Alexander P. Ivanov
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Correspondence to Alexander P. Ivanov.

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The authors declare that they have no conflicts of interest.

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MSC2010

70E40, 70E18, 37J60

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Borisov, A.V., Ivanov, A.P. A Top on a Vibrating Base: New Integrable Problem of Nonholonomic Mechanics. Regul. Chaot. Dyn. 27, 2–10 (2022). https://doi.org/10.1134/S1560354722010026

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  • DOI: https://doi.org/10.1134/S1560354722010026

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