Where has the monodromy gone in the 3-dim Champagne Bottle? (A case study in superintegrability)

Professor Francesco Fassò (University of Padova)

Frank Adams 2,

Monodromy is an obstruction to the triviality of the fibration by the invariant tori of an integrable or superintegrable Hamiltonian system; it was discovered by N.N. Nekhoroshev and by J.J. Duistermaat in the
1970s. The Champagne bottle (a point in a planar central force field shaped like the bottom of a wine bottle) is likely the simplest completely integrable system that exhibits monodromy (L. Bates, 1991). Like any planar central force field, the planar Champagne bottle is a subsystem of the system with the same central field, but in space. The spatial system is, actually, union of planar subsystems, and its invariant tori are those of the planar subsystems. The spatial system however cannot have monodromy, because the base of the fibration is simply connected.  In order to explain “where the monodromy went” in the 3-dim system (or, perhaps, “where it comes from” in the 2-dim subsystems) we describe the 3-dim system not as completely integrable, but as superintegrable, that is, in the realm of the theory developed by Nekhoroshev and Mischenko and Fomenko in the 1970s to describe systems which have more integrals of motion and more symmetry than what is needed for complete integrability. In superintegrable systems the invariant tori are isotropic, not Lagrangian, and there is also a
coarser, natural foliation of the phase space that is given by a Casimir map and has coisotropic leaves. The geometry of the regular part of these foliations is well understood (Nekhoroshev 1972, Dazord and Delzant 1987). It is the consideration of the singularities of the coisotropic structure that allows to understand the origin of the monodromy in the planar subsystems.

This is a join work with Larry Bates (Calgary).

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