Abstract:
We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.
Keywords:
nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure.
Citation:
Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere”, Regul. Chaotic Dyn., 16:5 (2011), 465–483
\Bibitem{BorKilMam11}
\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere
\jour Regul. Chaotic Dyn.
\yr 2011
\vol 16
\issue 5
\pages 465--483
\mathnet{http://mi.mathnet.ru/rcd447}
\crossref{https://doi.org/10.1134/S1560354711050042}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2844858}
\zmath{https://zbmath.org/?q=an:1309.37080}
Linking options:
https://www.mathnet.ru/eng/rcd447
https://www.mathnet.ru/eng/rcd/v16/i5/p465
This publication is cited in the following 26 articles:
Alexander A. Kilin, Elena N. Pivovarova, Tatiana B. Ivanova, “Rolling of a Homogeneous Ball on a Moving Cylinder”, Regul. Chaot. Dyn., 2024
Vladimir Dragović, Borislav Gajić, Bozidar Jovanović, “Spherical and Planar Ball Bearings — a Study of Integrable Cases”, Regul. Chaotic Dyn., 28:1 (2023), 62–77
Vladimir Dragović, Borislav Gajić, Bozidar Jovanović, “Spherical and Planar Ball Bearings — Nonholonomic Systems
with Invariant Measures”, Regul. Chaotic Dyn., 27:4 (2022), 424–442
Firdaus E. Udwadia, Nami Mogharabin, “New Directions in Modeling and Computational Methods for Complex Mechanical Dynamical Systems”, Processes, 10:8 (2022), 1560
I. S. Mamaev, A. A. Kilin, Yu. L. Karavaev, V. A. Shestakov, “Criteria of Motion Without Slipping for
an Omnidirectional Mobile Robot”, Rus. J. Nonlin. Dyn., 17:4 (2021), 527–546
Naprstek J. Fischer C., “Trajectories of a Ball Moving Inside a Spherical Cavity Using First Integrals of the Governing Nonlinear System”, Nonlinear Dyn., 106:3 (2021), 1591–1625
Antali M. Havas V. Hogan S.J. Stepan G., “Nonlinear Dynamics of a Basketball Rolling Around the Rim”, Nonlinear Dyn., 104:4 (2021), 3013–3037
Asiri S., Khademianzadeh F., Monadjemi A., Moallem P., “The Design and Development of a Dynamic Model of a Low-Power Consumption, Two-Pendulum Spherical Robot”, IEEE-ASME Trans. Mechatron., 24:5 (2019), 2406–2415
Hedrih Katica R. (Stevanovic), “Rolling Heavy Ball Over the Sphere in Real Rn3 Space”, Nonlinear Dyn., 97:1 (2019), 63–82
A. V. Borisov, A. A. Kilin, I. S. Mamaev, “O probleme Adamara–Gamelya i dinamike kolesnykh ekipazhei”, Nelineinaya dinam., 12:1 (2016), 145–163
Valery V. Kozlov, “The Dynamics of Systems with Servoconstraints. I”, Regul. Chaotic Dyn., 20:3 (2015), 205–224
Yury L. Karavaev, Alexander A. Kilin, “The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform”, Regul. Chaotic Dyn., 20:2 (2015), 134–152
Alexander A. Burov, Ekaterina S. Shalimova, “On the Motion of a Heavy Material Point on a Rotating Sphere (Dry Friction Case)”, Regul. Chaotic Dyn., 20:3 (2015), 225–233
Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204
A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Integral Yakobi v negolonomnoi mekhanike”, Nelineinaya dinam., 11:2 (2015), 377–396
A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “The Jacobi Integral in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:3 (2015), 383–400
Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “On the Hadamard–Hamel Problem and the Dynamics of Wheeled Vehicles”, Regul. Chaotic Dyn., 20:6 (2015), 752–766
A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Ierarkhiya dinamiki pri kachenii tverdogo tela bez proskalzyvaniya i vercheniya po ploskosti i sfere”, Nelineinaya dinam., 9:2 (2013), 141–202
S. V. Bolotin, T. V. Popova, “Ob uravneniyakh dvizheniya sistemy vnutri katyaschegosya shara”, Nelineinaya dinam., 9:1 (2013), 51–58
Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere”, Regul. Chaotic Dyn., 18:3 (2013), 277–328
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