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A Nonholonomic Model of the Paul Trap

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Abstract

In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincaré map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.

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Acknowledgments

The work of A.V. Borisov (Introduction, Section 1) was carried out at MIPT under project 5–100 for state support for leading universities of the Russian Federation. The work of A. A. Kilin (Sections 3, 5 and Appendix B) and I. S. Mamaev (Sections 2, 4 and Appendix A) was carried out within the framework of the state assignment to the Ministry of Education and Science of Russia (nos. 1.2404.2017/4.6 and 1.2405.2017/4.6, respectively).

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Authors and Affiliations

  1. A.A. Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334, Russia

    Alexey V. Borisov

  2. Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700, Russia

    Alexey V. Borisov

  3. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    Alexander A. Kilin

  4. Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069, Russia

    Ivan S. Mamaev

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  1. Alexey V. Borisov
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  2. Alexander A. Kilin
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  3. Ivan S. Mamaev
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Correspondence to Alexey V. Borisov.

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Borisov, A.V., Kilin, A.A. & Mamaev, I.S. A Nonholonomic Model of the Paul Trap. Regul. Chaot. Dyn. 23, 339–354 (2018). https://doi.org/10.1134/S1560354718030085

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  • DOI: https://doi.org/10.1134/S1560354718030085

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