A Nonholonomic Model of the Paul Trap

    2018, Volume 23, Number 3, pp.  339-354

    Author(s): Borisov A. V., Kilin A. A., Mamaev I. S.

    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 Poincar´e 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
    Citation: Borisov A. V., Kilin A. A., Mamaev I. S., A Nonholonomic Model of the Paul Trap, Regular and Chaotic Dynamics, 2018, Volume 23, Number 3, pp. 339-354

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