# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
 [1] D. Babbitt and V. Varadarajan,Local moduli for meromorphic differential equations. Sté. Math. France, Paris 1989. · Zbl 0683.34003 [2] L. Bates,Examples for obstructions to action-angle coordinates. Proc. Roy. Soc. Edinburgh,110, 27-30 (1988). · Zbl 0664.58004 [3] L. Bates and M. Zou,Degeneration and collapse of monodromy cycles (in preparation). · Zbl 0784.58026 [4] R. Cushman,Geometry of the energy-momentum mapping of the spherical pendulum. Centrum voor Wiskunde en Informatica Newsletter,1, 4-18 (1983). [5] R. Cushman and H. Knörrer,The Energy Momentum Mapping of the Lagrange Top. Lecture Notes in Math, Vol. 1139, pp. 12-24. Springer-Verlag, New York 1985. · Zbl 0615.70002 [6] J. J. Duistermaat,On global action-angle coordinates. Comm. Pure & appl. Math.,33, 687-706 (1980). · Zbl 0451.58016 · doi:10.1002/cpa.3160330602 [7] O. Forster,Lectures on Riemann Surfaces. Springer-Verlag, New York 1981. · Zbl 0475.30002 [8] E. Horozov,Perturbations of the spherical pendulum and Abelian integrals. J. reine. angew. Math.,408, 114-135 (1990). · Zbl 0692.58031 · doi:10.1515/crll.1990.408.114 [9] W. Poor,Differential Geometric Structures. McGraw-Hill, New York 1981. · Zbl 0493.53027 [10] J. C. van der Meer,The Hamiltonian Hopf bifurcation, Lect. Notes in Maths., Vol. 1160. Springer-Verlag, New York 1985. · Zbl 0585.58019 [11] A. Weinstein,Symplectic manifolds and their Lagrangian submanifolds. Adv. in Maths.,16, 329-346 (1971). · Zbl 0213.48203 · doi:10.1016/0001-8708(71)90020-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.