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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