Аннотация:
The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete constraints, we construct a discretization of the n-dimensional generalization of the Chaplygin sphere problem, which preserves the same first integrals as the continuous model, except the energy. We then study the discretization of the classical 3-dimensional problem for a class of special initial conditions, when an analog of the energy integral does exist and the corresponding map is given by an addition law on elliptic curves. The existence
of the invariant measure in this case is also discussed.
\RBibitem{Fed07}
\by Yuri N.~Fedorov
\paper A~Discretization of the Nonholonomic Chaplygin Sphere Problem
\jour SIGMA
\yr 2007
\vol 3
\papernumber 044
\totalpages 15
\mathnet{http://mi.mathnet.ru/sigma170}
\crossref{https://doi.org/10.3842/SIGMA.2007.044}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2299845}
\zmath{https://zbmath.org/?q=an:05241557}
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Эта публикация цитируется в следующих 8 статьяx:
Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367
Garcia-Naranjo L.C., Jimenez F., “The Geometric Discretisation of the Suslov Problem: a Case Study of Consistency For Nonholonomic Integrators”, Discret. Contin. Dyn. Syst., 37:8 (2017), 4249–4275
Luis C. garcía-Naranjo, Fernando Jiménez, “The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators”, Discrete & Continuous Dynamical Systems - A, 37:8 (2017), 4249
Iglesias-Ponte D., Carlos Marrero J., Martin de Diego D., Padron E., “Discrete Dynamics in Implicit Form”, Discret. Contin. Dyn. Syst., 33:3, SI (2013), 1117–1135
Jovanovic B., “Hamiltonization and Integrability of the Chaplygin Sphere in R-n”, J Nonlinear Sci, 20:5 (2010), 569–593
Jovanović B., “LR and L+R systems”, J. Phys. A, 42:22 (2009), 225202, 18 pp.
Iglesias D., Marrero J.C., de Diego D.M., Martínez E., “Discrete nonholonomic Lagrangian systems on Lie groupoids”, J. Nonlinear Sci., 18:3 (2008), 221–276
A.V. Borisov, Yu.N. Fedorov, I.S. Mamaev, “Chaplygin ball over a fixed sphere: an explicit integration”, Regul. Chaotic Dyn., 13:6 (2008), 557–571
Цитирование в формате AMSBIB
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