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December  2016, 8(4): 375-389. doi: 10.3934/jgm.2016012

No monodromy in the champagne bottle, or singularities of a superintegrable system

Larry M. Bates 1, and  Francesco Fassò 2,

1. 

Department of Mathematics, University of Calgary, Calgary, AB, T2N 1N4
2. 

Università di Padova, Dipartimento di Matematica Pura e Applicata, Via Trieste 63, 35121 Padova

Received  February 2015 Revised  August 2016 Published  November 2016

The three-dimensional champagne bottle system contains no mondromy, despite being entirely composed of invariant two-dimensional champagne bottle systems, each of which posesses nontrivial monodromy. We explain where the monodromy went in the three-dimensional system, or perhaps, where it did come from in the two-dimensional system, by regarding the three-dimensional system not as completely integrable, but as superintegrable (or non-commutatively integrable), and explaining the role of the singularities of its isotropic-coisotropic pair of foliations.
Keywords: superintegrability, symplectically complete isotropic foliations, Noncommutative integrability, Hamiltonian monodromy, singularities of integrable systems., complete integrability.
Mathematics Subject Classification: 70H06, 53D9.
Citation: Larry M. Bates, Francesco Fassò. No monodromy in the champagne bottle, or singularities of a superintegrable system. Journal of Geometric Mechanics, 2016, 8 (4) : 375-389. doi: 10.3934/jgm.2016012
References:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Graduate Text in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

L. Bates, Monodromy in the champagne bottle, Journal of Applied Mathematics and Physics (ZAMP), 42 (1991), 837-847. doi: 10.1007/BF00944566.  Google Scholar

[3]

R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, $2^{nd}$ edition, Birkhäuser, Basel, 2015. doi: 10.1007/978-3-0348-0918-4.  Google Scholar

[4]

P. Dazord and T. Delzant, Le probleme general des variables actions-angles, Journal of Differential Geometry, 26 (1987), 223-251.  Google Scholar

[5]

J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics, 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.  Google Scholar

[6]

H. Dullin and H. Hanßmann, The degenerate C. Neumann system I: symmetry reduction and convexity, Central European Journal of Mathematics, 10 (2012), 1627-1654. doi: 10.2478/s11533-012-0085-8.  Google Scholar

[7]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Applicandae Mathematicae, 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[8]

J. Milnor and J. Stasheff, Characteristic Classes, Annals of mathematics studies 76, Princeton University Press, 1974.  Google Scholar

[9]

A. Mischenko and A. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl, 12 (1978), 113-121. Google Scholar

[10]

N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moskow Math. Soc., 26 (1972), 181-198.  Google Scholar

[11]

M. Winnewisser, B. P. Winnewisser, F. C. De Lucia, I. R. Medvedev, S. C. Ross and L. M. Bates, The hidden kernel of molecular quasi-linearity: Quantum monodromy, Journal of Molecular Structure, 798 (2006), 1-26. doi: 10.1016/j.molstruc.2006.06.036.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Graduate Text in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

L. Bates, Monodromy in the champagne bottle, Journal of Applied Mathematics and Physics (ZAMP), 42 (1991), 837-847. doi: 10.1007/BF00944566.  Google Scholar

[3]

R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, $2^{nd}$ edition, Birkhäuser, Basel, 2015. doi: 10.1007/978-3-0348-0918-4.  Google Scholar

[4]

P. Dazord and T. Delzant, Le probleme general des variables actions-angles, Journal of Differential Geometry, 26 (1987), 223-251.  Google Scholar

[5]

J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics, 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.  Google Scholar

[6]

H. Dullin and H. Hanßmann, The degenerate C. Neumann system I: symmetry reduction and convexity, Central European Journal of Mathematics, 10 (2012), 1627-1654. doi: 10.2478/s11533-012-0085-8.  Google Scholar

[7]

F. Fassò, Superintegrable Hamiltonian systems: Geometry and perturbations, Acta Applicandae Mathematicae, 87 (2005), 93-121. doi: 10.1007/s10440-005-1139-8.  Google Scholar

[8]

J. Milnor and J. Stasheff, Characteristic Classes, Annals of mathematics studies 76, Princeton University Press, 1974.  Google Scholar

[9]

A. Mischenko and A. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl, 12 (1978), 113-121. Google Scholar

[10]

N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moskow Math. Soc., 26 (1972), 181-198.  Google Scholar

[11]

M. Winnewisser, B. P. Winnewisser, F. C. De Lucia, I. R. Medvedev, S. C. Ross and L. M. Bates, The hidden kernel of molecular quasi-linearity: Quantum monodromy, Journal of Molecular Structure, 798 (2006), 1-26. doi: 10.1016/j.molstruc.2006.06.036.  Google Scholar

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