Geodesic
On the Poisson structures for the Chaplygin ball and its generalizations
Russian journal of nonlinear dynamics, Tome 8 (2012) no. 2, pp. 345-353.

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Construction of the Poisson structures for the nonholonomic Chaplygin and Borisov–Mamaev–Fedorov systems is discussed. The corresponding vector fields are conformally Hamiltonian and generalized conformally Hamiltonian vector fields with respect to the linear in momenta Poisson brackets. We suppose that this difference is closely related with the non-trivial deformation of canonical Poisson bivector, which appears in the Borisov–Mamaev–Fedorov case.
Keywords: nonholonomic mechanics, Chaplygin sphere
Mots-clés : Poisson brackets. 
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A. V. Tsiganov. On the Poisson structures for the Chaplygin ball and its generalizations. Russian journal of nonlinear dynamics, Tome 8 (2012) no. 2, pp. 345-353. http://geodesic.mathdoc.fr/item/ND_2012_8_2_a8/

[1] Borisov A. V., Fëdorov Yu. N., “O dvukh vidoizmenennykh integriruemykh zadachakh dinamiki”, Vestn. Mosk. un-ta. Ser. 1. Matem. Mekhan., 1995, no. 6, 102–105 ; Неголономные динамические системы: Интегрируемость, хаос, странные аттракторы, Сб. ст., ред. А. В. Борисов, И. С. Мамаев, Институт компьютерных исследований, М.–Ижевск, 2002, 67–70 | MR | Zbl

[2] Borisov A. V., Mamaev I. S., “Gamiltonovost zadachi Chaplygina o kachenii shara”, Matem. zametki, 70:5 (2001), 793–795 | MR

[3] Borisov A. V., Mamaev I. S., Marikhin V. G., “Yavnoe integrirovanie odnoi negolonomnoi zadachi”, Dokl. RAN, 422:4 (2008), 475–478 | MR | Zbl

[4] Borisov A. V., Mamaev I. S., “Zakony sokhraneniya, ierarkhiya dinamiki i yavnoe integrirovanie negolonomnykh sistem”, Nelineinaya dinamika, 4:3 (2008), 223–280

[5] Borisov A. V., Fedorov Yu. N., Mamaev I. S., “Chaplygin ball over a fixed sphere: An explicit integration”, Regul. Chaotic Dyn., 13:6 (2008), 557–571 | DOI | MR | Zbl

[6] Borisov A. V., Kilin A. A., Mamaev I. S., “Obobschenie preobrazovaniya Chaplygina i yavnoe integrirovanie sharovogo podvesa”, Nelineinaya dinamika, 7:2 (2011), 313–338 | MR

[7] Chaplygin S. A., “O katanii shara po gorizontalnoi ploskosti”, Matem. sb., 24 (1903), 139–168 ; Чаплыгин С. А., Собр. соч., т. 1, ОГИЗ, М.–Л., 1948, 76–101

[8] Tsiganov A. V., “Integrable Euler top and nonholonomic Chaplygin ball”, J. Geomet. Mech., 3:3 (2011), 337–362 | DOI | MR | Zbl

[9] Tsiganov A. V., “One invariant measure and different Poisson brackets for two non-holonomic systems”, Regul. Chaotic Dyn., 17:1 (2012), 72–96 | DOI | MR

[10] Tsyganov A. V., “O negolonomnykh sistemakh Veselovoi i Chaplygina”, Nelineinaya dinamika, 2012 (to appear)

[11] Turiel F., “Structures bihamiltoniennes sur le fibré cotangent”, C. R. Acad. Sci. Paris Sér. 1 Math., 315 (1992), 1085–1088 | MR | Zbl