Аннотация:
We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu.N.Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support – the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.
This research was supported by the Grant of the Government of the Russian Federation
for state support of scientific research conducted under supervision of leading scientists in
Russian educational institutions of higher professional education (contract no. 11.G34.31.0039)
and the Federal target programme “Scientific and Scientific-Pedagogical Personnel of Innovative
Russia”, measure 1.1. “Scientific-Educational Center Regular and Chaotic Dynamics” (project
code 02.740.11.0195), measure 1.5 “Topology and Mechanics” (project code 14.740.11.0876). The
work of A. A.Kilin was supported by the Grant of the President of the Russian Federation for the
Support of Young Russian Scientists–Candidates of Science (MK-8428.2010.1).
Поступила в редакцию: 27.07.2011 Принята в печать: 19.11.2011
Образец цитирования:
Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support”, Regul. Chaotic Dyn., 17:2 (2012), 170–190
\RBibitem{BorKilMam12}
\by Alexey V.~Borisov, Alexander A.~Kilin, Ivan S.~Mamaev
\paper Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support
\jour Regul. Chaotic Dyn.
\yr 2012
\vol 17
\issue 2
\pages 170--190
\mathnet{http://mi.mathnet.ru/rcd338}
\crossref{https://doi.org/10.1134/S1560354712020062}
\zmath{https://zbmath.org/?q=an:1253.37063}
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd338
https://www.mathnet.ru/rus/rcd/v17/i2/p170
Эта публикация цитируется в следующих 26 статьяx:
Alexander A. Kilin, Elena N. Pivovarova, “Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane”, Nonlinear Dyn, 2024
A. A. Kilin, T. B. Ivanova, “The Integrable Problem of the Rolling Motion
of a Dynamically Symmetric Spherical Top
with One Nonholonomic Constraint”, Rus. J. Nonlin. Dyn., 19:1 (2023), 3–17
Alexander A. Kilin, Elena N. Pivovarova, “A Particular Integrable Case in the Nonautonomous Problem
of a Chaplygin Sphere Rolling on a Vibrating Plane”, Regul. Chaotic Dyn., 26:6 (2021), 775–786
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
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
Bizyaev I.A. Borisov A.V. Mamaev I.S., “Dynamics of the Chaplygin Ball on a Rotating Plane”, Russ. J. Math. Phys., 25:4 (2018), 423–433
Sokolov S.V. Ryabov P.E., “Bifurcation Analysis of the Dynamics of Two Vortices in a Bose–Einstein Condensate. the Case of Intensities of Opposite Signs”, Regul. Chaotic Dyn., 22:8 (2017), 976–995
Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367
А. В. Борисов, И. С. Мамаев, И. А. Бизяев, “Динамические системы с неинтегрируемыми связями: вакономная механика, субриманова геометрия и неголономная механика”, УМН, 72:5(437) (2017), 3–62; A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840
Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248
А. В. Борисов, А. А. Килин, И. С. Мамаев, “О проблеме Адамара–Гамеля и динамике колесных экипажей”, Нелинейная динам., 12:1 (2016), 145–163
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
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
Bolsinov A.V., Kilin A.A., Kazakov A.O., “Topological Monodromy as An Obstruction to Hamiltonization of Nonholonomic Systems: Pro Or Contra?”, J. Geom. Phys., 87 (2015), 61–75
Rosemann S., Schoebel K., “Open Problems in the Theory of Finite-Dimensional Integrable Systems and Related Fields”, J. Geom. Phys., 87 (2015), 396–414
Ю. Л. Караваев, А. А. Килин, “Динамика сфероробота с внутренней омниколесной платформой”, Нелинейная динам., 11:1 (2015), 187–204
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
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
Alexey V. Borisov, Ivan S. Mamaev, “Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere”, Regul. Chaotic Dyn., 18:4 (2013), 356–371
Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “How to Control the Chaplygin Ball Using Rotors. II”, Regul. Chaotic Dyn., 18:1-2 (2013), 144–158
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