Abstract:
Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.
Citation:
A. V. Borisov, I. S. Mamaev, “Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting”, Regul. Chaotic Dyn., 12:2 (2007), 153–159
\Bibitem{BorMam07}
\by A. V. Borisov, I. S. Mamaev
\paper Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting
\jour Regul. Chaotic Dyn.
\yr 2007
\vol 12
\issue 2
\pages 153--159
\mathnet{http://mi.mathnet.ru/rcd618}
\crossref{https://doi.org/10.1134/S1560354707020037}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2350303}
\zmath{https://zbmath.org/?q=an:1229.37081}
Linking options:
https://www.mathnet.ru/eng/rcd618
https://www.mathnet.ru/eng/rcd/v12/i2/p153
This publication is cited in the following 32 articles:
Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Gyroscopic Chaplygin Systems and Integrable Magnetic Flows on Spheres”, J Nonlinear Sci, 33:3 (2023)
Aleksandar Obradović, Zoran Mitrović, Slaviša Šalinić, “On the problem of a heavy homogeneous ball rolling without slipping over a fixed surface of revolution”, Applied Mathematics and Computation, 420 (2022), 126906
Firdaus E. Udwadia, Nami Mogharabin, “New Directions in Modeling and Computational Methods for Complex Mechanical Dynamical Systems”, Processes, 10:8 (2022), 1560
Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis”, Theor. Appl. Mech., 47:2 (2020), 257–287
B. Gajić, B. Jovanović, “Two Integrable Cases of a Ball Rolling over a Sphere in Rn”, Rus. J. Nonlin. Dyn., 15:4 (2019), 457–475
Božidar Jovanović, “Note on a ball rolling over a sphere: integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization”, Theor. Appl. Mech., 46:1 (2019), 97–108
Gajic B. Jovanovic B., “Nonholonomic Connections, Time Reparametrizations, and Integrability of the Rolling Ball Over a Sphere”, Nonlinearity, 32:5 (2019), 1675–1694
Božidar Jovanović, “Rolling balls over spheres in Rn”, Nonlinearity, 31:9 (2018), 4006
Valery Kozlov, “The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems”, J. Dyn. Control Syst., 22:4 (2016), 713–724
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
Božidar Jovanović, “Invariant Measures of Modified LR and L+R Systems”, Regul. Chaotic Dyn., 20:5 (2015), 542–552
Alexander P. Ivanov, “On the Control of a Robot Ball Using Two Omniwheels”, Regul. Chaotic Dyn., 20:4 (2015), 441–448
Alexey V. Borisov, Ivan S. Mamaev, Alexander A. Kilin, Ivan A. Bizyaev, “Qualitative Analysis of the Dynamics of a Wheeled Vehicle”, Regul. Chaotic Dyn., 20:6 (2015), 739–751
Alexander A. Kilin, Elena N. Pivovarova, Tatyana B. Ivanova, “Spherical Robot of Combined Type: Dynamics and Control”, Regul. Chaotic Dyn., 20:6 (2015), 716–728
Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204
E. N. Pivovarova, A. V. Klekovkin, “Vliyanie treniya kacheniya na upravlyaemoe dvizhenie robota-kolesa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:4 (2015), 583–592
A. A. Kilin, Yu. L. Karavaev, A. V. Klekovkin, “Kinematicheskaya model upravleniya vysokomanevrennym mobilnym sferorobotom s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 10:1 (2014), 113–126
A. A. Kilin, Yu. L. Karavaev, “Kinematicheskaya model upravleniya sferorobotom s neuravnoveshennoi omnikolesnoi platformoi”, Nelineinaya dinam., 10:4 (2014), 497–511
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
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
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