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
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.
Paul trap, stability, nonholonomic system, three-dimensional map, gyroscopic stabilization, noninertial coordinate system, Poincaré map, nonholonomic constraint, rolling without slipping, region of linear stability
|Ministry of Education and Science of the Russian Federation
|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).
MSC: 37J60, 34A34
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
\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper A Nonholonomic Model of the Paul Trap
\jour Regul. Chaotic Dyn.
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