Regular and Chaotic Dynamics

, Volume 23, Issue 3, pp 339–354 | Cite as

A Nonholonomic Model of the Paul Trap

  • Alexey V. Borisov
  • Alexander A. Kilin
  • Ivan S. Mamaev


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 

MSC2010 numbers

37J60 34A34 


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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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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
  • Alexander A. Kilin
    • 3
  • Ivan S. Mamaev
    • 4
  1. 1.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  3. 3.Udmurt State UniversityIzhevskRussia
  4. 4.Izhevsk State Technical UniversityIzhevskRussia

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