Аннотация:
In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
This research was done at the Udmurt State University and was supported by the Grant Program
of the Government of the Russian Federation for state support of scientific research conducted
under the supervision of leading scientists at Russian institutions of higher professional education
(Contract No11.G34.31.0039).
Поступила в редакцию: 04.08.2012 Принята в печать: 19.10.2012
Образец цитирования:
Alexey V. Bolsinov, Alexey V. Borisov, Ivan S. Mamaev, “Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals”, Regul. Chaotic Dyn., 17:6 (2012), 571–579
\RBibitem{BolBorMam12}
\by Alexey V.~Bolsinov, Alexey V.~Borisov, Ivan S.~Mamaev
\paper Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals
\jour Regul. Chaotic Dyn.
\yr 2012
\vol 17
\issue 6
\pages 571--579
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\crossref{https://doi.org/10.1134/S1560354712060081}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3001102}
\zmath{https://zbmath.org/?q=an:1267.37066}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2012RCD....17..571B}
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https://www.mathnet.ru/rus/rcd350
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Эта публикация цитируется в следующих 37 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
G. R. Saypulaev, B. I. Adamov, A. I. Kobrin, “Comparative Analysis of the Dynamics of a Spherical
Robot with a Balanced Internal Platform Taking into
Account Different Models of Contact Friction”, Rus. J. Nonlin. Dyn., 18:5 (2022), 803–815
Alexey Mashtakov, 2021 International Conference “Nonlinearity, Information and Robotics” (NIR), 2021, 1
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
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
Borisov A. Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520
Jovanovic B., “Rolling Balls Over Spheres in R-N”, Nonlinearity, 31:9 (2018), 4006–4030
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
А. В. Борисов, А. О. Казаков, Е. Н. Пивоварова, “Регулярная и хаотическая динамика в «резиновой» модели волчка Чаплыгина”, Нелинейная динам., 13:2 (2017), 277–297
И. А. Бизяев, А. В. Борисов, И. С. Мамаев, “Случай Гесса–Аппельрота и квантование числа вращения”, Нелинейная динам., 13:3 (2017), 433–452
И. А. Бизяев, А. В. Борисов, И. С. Мамаев, “Система Гесса–Аппельрота и ее неголономные аналоги”, Современные проблемы математики, механики и математической физики. II, Сборник статей, Труды МИАН, 294, МАИК «Наука/Интерпериодика», М., 2016, 268–292; I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “The Hess–Appelrot system and its nonholonomic analogs”, Proc. Steklov Inst. Math., 294 (2016), 252–275
Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248
И. А. Бизяев, А. В. Борисов, А. О. Казаков, “Динамика задачи Суслова в поле тяжести: реверс и странные аттракторы”, Нелинейная динам., 12:2 (2016), 263–287
Alexey V. Borisov, Alexey O. Kazakov, Elena N. Pivovarova, “Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 885–901
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
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
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