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Monodromy in the champagne bottle. (English) Zbl 0755.58028
Let \(F: M\to B\) be a fibration whose fibers are Lagrangian tori. The problem of the existence of action-angle variables has been studied by J. J. Duistermaat [Commun. Pure Appl. Math. 33, No. 6, 687-706 (1980; Zbl 0439.58014)]. There are two obstructions to the global existence of such variables: the Chern class of the fibration and the monodromy of a bundle.
In a previous paper [Proc. R. Soc. Edinb., Sect. A 110, No. 1/2, 27-30 (1988; Zbl 0664.58004)] the author builds examples in which the monodromy is trivial but the Chern class is not. In this paper an example with nontrivial monodromy is built: the Hamiltonian description of a particle moving in a potential field shaped like the punt of a champagne bottle. Other examples with nontrivial monodromy have already been observed in the literature, for instance, the spherical pendulum, the Langrangian top or the Hamiltonian Hopf bifurcation. The relevance of this example lies in its simplicity because there is no complicated topology in the phase space. An exposition of monodromy can be found in an appendix.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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