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Regul. Chaotic Dyn., 2018, Volume 23, Issue 3, Pages 339–354 (Mi rcd327)  

A Nonholonomic Model of the Paul Trap

Alexey V. Borisovab, Alexander A. Kilinc, Ivan S. Mamaevd

a Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia
b A. A. Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334 Russia
c Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
d Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069 Russia

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 balls 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.

Keywords: Paul trap, stability, nonholonomic system, three-dimensional map, gyroscopic stabilization, noninertial coordinate system, Poincaré map, nonholonomic constraint, rolling without slipping, region of linear stability

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation
1.2404.2017/4.6
1.2405.2017/4.6
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).


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

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Document Type: Article
MSC: 37J60, 34A34
Received: 12.03.2018
Language: English

Citation: Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “A Nonholonomic Model of the Paul Trap”, Regul. Chaotic Dyn., 23:3 (2018), 339–354

Citation in format AMSBIB
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\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper A Nonholonomic Model of the Paul Trap
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 3
\pages 339--354
\mathnet{http://mi.mathnet.ru/rcd327}
\crossref{https://doi.org/10.1134/S1560354718030085}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3811823}
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