On the Poisson structures for the nonholonomic Chaplygin and Veselova problems
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 \(e(3)^*\). Here, \(\omega\) is the angular velocity and \(M\) is the angular momentum of the ball with respect to the contact point.
scientific article; zbMATH DE number 6104793
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