Abstract:
We discuss a Poisson structure, linear in momenta, for the generalized nonholonomic Chaplygin sphere problem and the $LR$ Veselova system. Reduction of these structures to the canonical form allows one to prove that the Veselova system is equivalent to the Chaplygin ball after transformations of coordinates and parameters.
The work was done at the Udmurt State University and was supported by the Grant Program of the Government of the Russian Federation for Support for the Scientific Research Project implemented under the supervision of leading scientists at Russian institutions of higher education (11.G34.31.0039).
Citation:
Andrey V. Tsiganov, “On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems”, Regul. Chaotic Dyn., 17:5 (2012), 439–450
\Bibitem{Tsi12}
\by Andrey V. Tsiganov
\paper On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems
\jour Regul. Chaotic Dyn.
\yr 2012
\vol 17
\issue 5
\pages 439--450
\mathnet{http://mi.mathnet.ru/rcd414}
\crossref{https://doi.org/10.1134/S1560354712050061}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2989516}
\zmath{https://zbmath.org/?q=an:1263.37074}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2012RCD....17..439T}
Linking options:
https://www.mathnet.ru/eng/rcd414
https://www.mathnet.ru/eng/rcd/v17/i5/p439
This publication is cited in the following 16 articles:
Andrey V. Tsiganov, “Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane”, Regul. Chaotic Dyn., 24:2 (2019), 171–186
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
Jovanovic B., “Rolling Balls Over Spheres in R-N”, Nonlinearity, 31:9 (2018), 4006–4030
Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179
Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.
Andrey V. Tsiganov, “On Integrable Perturbations of Some Nonholonomic Systems”, SIGMA, 11 (2015), 085, 19 pp.
A. V. Bolsinov, A. A. Kilin, A. O. Kazakov, “Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?”, J. Geom. Phys., 87 (2015), 61–75
A. V. Borisov, I. S. Mamaev, A. V. Tsiganov, “Non-holonomic dynamics and Poisson geometry”, Russian Math. Surveys, 69:3 (2014), 481–538
Andrey V. Tsiganov, “On the Lie Integrability Theorem for the Chaplygin Ball”, Regul. Chaotic Dyn., 19:2 (2014), 185–197
Andrey Tsiganov, “Poisson structures for two nonholonomic systems with partially reduced symmetries”, Journal of Geometric Mechanics, 6:3 (2014), 417
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
A. V. Bolsinov, A. A. Kilin, A. O. Kazakov, “Topologicheskaya monodromiya v negolonomnykh sistemakh”, Nelineinaya dinam., 9:2 (2013), 203–227
A. V. Tsyganov, “O share Chaplygina v absolyutnom prostranstve”, Nelineinaya dinam., 9:4 (2013), 711–719
Andrey V. Tsiganov, “On a Trivial Family of Noncommutative Integrable Systems”, SIGMA, 9 (2013), 015, 13 pp.
I A Bizyaev, A V Tsiganov, “On the Routh sphere problem”, J. Phys. A: Math. Theor., 46:8 (2013), 085202
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