Geodesic
On stability of the Chaplygin ball motion on a plane with an arbitrary friction law
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2012), pp. 140-145
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The Chaplygin ball on a plane is considered under the action of the friction force which satisfies the following condition: $(\mathbf F,\mathbf u)<0$ as $\mathbf u\neq0$ and $\mathbf F=0$ as $\mathbf u=0$, where $\mathbf u$ is the gliding velocity. The ball is supposed to have a point contact with the supporting plane (this means that the contact spot is absent and also there is no rotation friction torque). The main task of the paper is to determine a set of possible stationary (or final) motions and their stability. In the current paper it is shown that exactly three stationary motions are possible; these motions represent straightline uniform rolling motions of the ball without sliding, at that the ball is rotating around one of the primary axes of the inertia tensor. Rotation around the axis of the greatest moment of inertia is stable, around the middle one and the lowest one it is unstable.
Keywords: Chaplygin ball, stationary motions, stability. 
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I. I. Ovsyannikov. On stability of the Chaplygin ball motion on a plane with an arbitrary friction law. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2012), pp. 140-145. http://geodesic.mathdoc.fr/item/VUU_2012_4_a10/

[1] Chaplygin S. A., “O katanii shara po gorizontalnoi ploskosti”, Matematicheskii sbornik, 24:1 (1903), 139–168 | Zbl

[2] Gallop E. G., “On the rise of a spinning top”, Proc. Cambr. Phylos. Soc., 19, Part 3 (1904), 356–373 | Zbl

[3] Kilin A. A., “The dynamics of Chaplygin ball: the qualitative and computer analysis”, Regular and Chaotic Dynamics, 6:3 (2001), 291–306 | DOI | MR | Zbl

[4] Karapetyan A. V., Ustoichivost statsionarnykh dvizhenii, Editorial URSS, M., 1998, 166 pp.

[5] Moschuk N. K., “O dvizhenii shara Chaplygina na gorizontalnoi ploskosti”, Prikl. mat. i mekh., 47:6 (1983), 916–921

[6] Moschuk N. K., “O dvizhenii tyazhelogo tverdogo tela na gorizontalnoi ploskosti s vyazkim treniem”, Prikl. mat. i mekh., 49:1 (1985), 66–71

[7] Karapetyan A. V., “Globalnyi kachestvennyi analiz dinamiki kitaiskogo volchka (tip-top)”, Izvestiya RAN. Mekhanika tverdogo tela, 2008, no. 3, 33–41 | MR

[8] Markeev A. P., Dinamika tela, soprikasayuschegosya s tverdoi poverkhnostyu, Nauka, M., 1992, 336 pp. ; 2-е издание, Регулярная и хаотическая динамика, Ижевск, 2011 | MR | Zbl