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The energy momentum mapping of the Lagrange top. (English) Zbl 0615.70002
Differential geometric methods in mathematical physics, Proc. Int. Conf., Clausthal/Ger. 1983, Lect. Notes Math. 1139, 12-24 (1985).
[For the entire collection see Zbl 0564.00012.]
The motion of a symmetric top about a fixed point is analyzed within the modern framework of Hamiltonian systems on Lie groups. The authors first recall the complete integrability of the system by an explicit descripion of an \({\mathbb{R}}^ 3\)-valued energy-momentum mapping.
Using the Marsden-Weinstein reduction process, a one-degree of freedom Hamiltonian system is constructed on a two-dimensional topological manifold. For regular values of the energy-momentum map, the leaves of the corresponding fibration are shown to be diffeomorphic to \(T^ 3\). In the main section of the paper, a detailed analysis is given of the set of critical values, which is found to have ”the shape of a bowl with a thread joining opposite sides”. Finally, the energy momentum mapping is shown to have monodromy, meaning that the inverse image of the set of regular values is a nontrivial fibre bundle.
Reviewer: W.Sarlet

MSC:
70E05 Motion of the gyroscope
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems