Skip to main content
Log in

Monodromy in the champagne bottle

  • Original Papers
  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Abstract

We show that the Hamiltonian description of a particle moving in a potential field shaped like the punt of a champagne bottle (more properly anS 1 symmetric double well) has monodromy, which is a global obstruction to the construction of action-angle variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. D. Babbitt and V. Varadarajan,Local moduli for meromorphic differential equations. Sté. Math. France, Paris 1989.

    Google Scholar 

  2. L. Bates,Examples for obstructions to action-angle coordinates. Proc. Roy. Soc. Edinburgh,110, 27–30 (1988).

    Google Scholar 

  3. L. Bates and M. Zou,Degeneration and collapse of monodromy cycles (in preparation).

  4. R. Cushman,Geometry of the energy-momentum mapping of the spherical pendulum. Centrum voor Wiskunde en Informatica Newsletter,1, 4–18 (1983).

    Google Scholar 

  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.

    Google Scholar 

  6. J. J. Duistermaat,On global action-angle coordinates. Comm. Pure & appl. Math.,33, 687–706 (1980).

    Google Scholar 

  7. O. Forster,Lectures on Riemann Surfaces. Springer-Verlag, New York 1981.

    Google Scholar 

  8. E. Horozov,Perturbations of the spherical pendulum and Abelian integrals. J. reine. angew. Math.,408, 114–135 (1990).

    Google Scholar 

  9. W. Poor,Differential Geometric Structures. McGraw-Hill, New York 1981.

    Google Scholar 

  10. J. C. van der Meer,The Hamiltonian Hopf bifurcation, Lect. Notes in Maths., Vol. 1160. Springer-Verlag, New York 1985.

    Google Scholar 

  11. A. Weinstein,Symplectic manifolds and their Lagrangian submanifolds. Adv. in Maths.,16, 329–346 (1971).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Partially supported by NSERC grant OGP0042416.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bates, L.M. Monodromy in the champagne bottle. Z. angew. Math. Phys. 42, 837–847 (1991). https://doi.org/10.1007/BF00944566

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00944566

Keywords

Navigation