Abstract:
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
This research was supported by the Target Programmes for 2012–2014 (State contract 1.1248.2011, 1.7734.2013) and grant RFBR 13-01-12462-ofi_m. A. A. Kilin’s research was supported by the grant of the President of the Russian Federation for the Support of Young Russian Scientists–Doctors of Science (MD-2324.2013.1).
Citation:
Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Problem of Drift and Recurrence for the Rolling Chaplygin Ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859
\Bibitem{BorKilMam13}
\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper The Problem of Drift and Recurrence for the Rolling Chaplygin Ball
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 6
\pages 832--859
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Ivan A. Bizyaev, Ivan S. Mamaev, “Roller Racer with Varying Gyrostatic Momentum:
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E. A. Mikishanina, “Nonholonomic mechanical systems on a plane with a variable slope”, Zhurnal SVMO, 25:4 (2023), 326–341
E. A. Mikishanina, “Rolling motion dynamics of a spherical robot with a pendulum actuator controlled by the Bilimovich servo-constraint”, Theoret. and Math. Phys., 211:2 (2022), 679–691
E. A. Mikishanina, “Motion Control of a Spherical Robot with a Pendulum
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Alexander A. Kilin, Elena N. Pivovarova, “Motion control of the spherical robot rolling on a vibrating plane”, Applied Mathematical Modelling, 109 (2022), 492
Alexander A. Kilin, Elena N. Pivovarova, “A Particular Integrable Case in the Nonautonomous Problem
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Alexey V. Borisov, Evgeniya A. Mikishanina, “Two Nonholonomic Chaotic Systems. Part II. On the Rolling of a Nonholonomic Bundle of Two Bodies”, Regul. Chaotic Dyn., 25:4 (2020), 392–400
A. V. Borisov, E. A. Mikishanina, “Dynamics of the Chaplygin Ball with Variable Parameters”, Rus. J. Nonlin. Dyn., 16:3 (2020), 453–462
Bizyaev I.A. Mamaev I.S., “Separatrix Splitting and Nonintegrability in the Nonholonomic Rolling of a Generalized Chaplygin Sphere”, Int. J. Non-Linear Mech., 126 (2020), 103550
Putkaradze V. Rogers S., “on the Optimal Control of a Rolling Ball Robot Actuated By Internal Point Masses”, J. Dyn. Syst. Meas. Control-Trans. ASME, 142:5 (2020)
Alexander A. Kilin, Elena N. Pivovarova, “Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 24:2 (2019), 212–233
Ivan Yu. Polekhin, “Precession of the Kovalevskaya and Goryachev – Chaplygin Tops”, Regul. Chaotic Dyn., 24:3 (2019), 281–297
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem”, Regul. Chaotic Dyn., 24:5 (2019), 560–582
Borisov A. Kilin A. Mamaev I., “Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard”, Int. J. Bifurcation Chaos, 29:3 (2019), 1930008
Alexander A. Kilin, Elena N. Pivovarova, “Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 23:7-8 (2018), 887–907
V. Putkaradze, S. Rogers, “On the dynamics of a rolling ball actuated by internal point masses”, Meccanica, 53:15 (2018), 3839–3868
I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dynamics of the Chaplygin ball on a rotating plane”, Russ. J. Math. Phys., 25:4 (2018), 423–433
Alexander A. Kilin, Elena N. Pivovarova, “The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane”, Regul. Chaotic Dyn., 22:3 (2017), 298–317
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