A Nonholonomic Model of the Paul Trap
- 56 Downloads
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.
KeywordsPaul trap stability nonholonomic system three-dimensional map gyroscopic stabilization noninertial coordinate system Poincaré map nonholonomic constraint rolling without slipping region of linear stability
MSC2010 numbers37J60 34A34
Unable to display preview. Download preview PDF.
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).
- 1.Arnol’d, V. I., Kozlov, V.V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.Google Scholar
- 8.Brouwer, L. E. J., Collected Works: Vol. 2, H. Freudenthal (Ed.), Amsterdam: North-Holland, 1976.Google Scholar
- 9.Brouwer, L.E. J., The Motion of a Particle on the Bottom of a Rotating Vessel under the Influence of the Gravitational Force, in Collected Works: Vol. 2, H. Freudenthal (Ed.), Amsterdam: North-Holland, 1976, pp. 665–686.Google Scholar
- 16.Meiss, J. D., Miguel, N., Simó, C., and Vieiro, A., Accelerator Modes and Anomalous Diffusion in 3D Volume-Preserving Maps, arXiv:1802.10484 (2018).Google Scholar
- 24.Routh, E. J., Dynamics of a System of Rigid Bodies: Elementary Part. Being Part 1 of a Treatise on the Whole Subject, 7th ed., rev. and enl., New York: Dover, 1960.Google Scholar