On the Poisson structures for the nonholonomic Chaplygin and Veselova problems
type: Item
instance of: scholarly article
title: On the Poisson structures for the nonholonomic Chaplygin and Veselova problems
zbMATH Open document ID: 1263.37074
DOI: 10.1134/S1560354712050061
published in: Regular and Chaotic Dynamics
publication date: 2012-11-12T00:00:00Z
review text: The author studies three well-known integrable nonholonomic rigid body problems with an invariant measure: the classical Chaplygin sphere problem [\textit{S. Tschaplygin}, Moskau. Math. Samml. 24, 139-168 (1903; JFM 34.0782.01) (Russian), English translation: Regul. Chaotic Dyn. 7, No. 2, 131-148 (2002; Zbl 1058.70003)], the Veselova problem [\textit{A. Veselov} and \textit{L. E. Veselova,} Math. Notes 44, No. 5, 810--819 (1988); translation from Mat. Zametki 44, No. 5, 604--619 (1988; Zbl 0694.58021)] and the Borisov-Fedorov generalization of the Chaplygin sphere problem (rolling of a dynamically asymmetric ball over a fixed sphere) [\textit{A. B. Borisov} and \textit{Yu. N. Fedorov}; Mosc. Univ. Mech. Bull. 50, No. 6, 16--18 (1995); translation from Vestn. Mosk. Univ., Ser. I 1995, No. 6, 102--105 (1995; Zbl 0925.70097); \textit{A. B. Borisov, Yu. N. Fedorov} and \textit{I. S. Mamaev}, Regul. Chaotic Dyn. 13, No. 6, 557--571 (2008; Zbl 1229.37080)]. Firstly, the interpretation of Fedorov's relationship between Veselova and Chaplygin problems [\textit{Yu. N. Fedorov}, Mosc. Univ. Mech. Bull. 44, No. 4, 7--12 (1989); translation from Vestn. Mosk. Univ., Ser. I 1989, No. 4, 38--41 (1989; Zbl 0703.70015)] is given in terms of the Poisson mapping between appropriate Hamiltonizations of the systems; see [\textit{A. B. Borisov} and \textit{I. S. Mamaev}, Sib. Mat. Zh. 48, No. 1, 33--45 (2007); translation in Sib. Math. J. 48, No. 1, 26--36 (2007; Zbl 1164.37342)]. Secondly, concerning the Borisov-Fedorov generalization of the Chaplygin sphere problem, the following interesting result is proved: The initial non-Hamiltonian vector field \(X\) can be expressed as a sum of two conformally Hamiltonian vector fields \(X=g_1^{-1} X_1+ g_2^{-1} X_2\), where \(X_1\) and \(X_2\) are Hamiltonian vector fields associated with two commuting integrals of motion \(H_1=(M,\omega)\), \(H_2=(M,M)\) with respect to a certain non-trivial transformation of the Lie-Poisson bracket on \(e(3)^*…
Mathematics Subject Classification ID: 37J60
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 70H45
zbMATH DE Number: 6104793
zbMATH Keywords: nonholonomic systems
zbMATH Keywords: integrability
zbMATH Keywords: Poisson bracket
zbMATH Keywords: Chaplygin reducing multiplier
author: A. V. Tsiganov
MaRDI profile type: Publication
cites work: Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems
cites work: Chaplygin ball over a fixed sphere: an explicit integration
cites work: Generalized Chaplygin's transformation and explicit integration of a system with a spherical support
cites work: Nonholonomic Lagrangian systems on Lie algebroids
cites work: Les variétés de Poisson et leurs algèbres de Lie associees
cites work: Q3785520
cites work: Transformation of a pair of commuting hamiltonians quadratic in momenta to canonical form and real partial separation of variables for the Clebsch top
cites work: On two different bi-Hamiltonian structures for the Toda lattice
cites work: Integrable Euler top and nonholonomic Chaplygin ball
cites work: One invariant measure and different Poisson brackets for two non-holonomic systems
cites work: Q4031263
cites work: Q3811394
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description: scientific article; zbMATH DE number 6104793