A Nonholonomic Model of the Paul Trap |
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Authors: | Alexey V. Borisov Alexander A. Kilin Ivan S. Mamaev |
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Affiliation: | 1.A.A. Blagonravov Mechanical Engineering Research Institute of RAS,Moscow,Russia;2.Moscow Institute of Physics and Technology,Dolgoprudnyi,Russia;3.Udmurt State University,Izhevsk,Russia;4.Izhevsk State Technical University,Izhevsk,Russia |
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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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