Communications in Mathematical Physics

, Volume 120, Issue 2, pp 269–294 | Cite as

The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case

  • R. Montgomery
Article

Abstract

We show how averaging defines an Ehresmann connection whose holonomy is the classical adiabatic angles which Hannay defined for families of completely integrable systems. The averaging formula we obtain for the connection only requires that the family of Hamiltonians has a continuous symmetry group. This allows us to extend the notion of the Hannay angles to families of non-integrable systems with symmetry. We state three geometric axioms satisfied by the connection. These axioms uniquely determine the connection, thus enabling us to find new formulas for the connection and its curvature. Two examples are given.

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References

  1. Anandan, J.: Geometric angles in quantum and classical physics. Phys. Lett. A (in press 1988)Google Scholar
  2. Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull Lond. Math. Soc.14, 1–15 (1982)Google Scholar
  3. Arnold, V.I.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  4. Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations. Berlin, Heidelberg, New York: Springer 1983Google Scholar
  5. Berry, M.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A392, 45 (1984)Google Scholar
  6. Berry, M.: Classical adiabatic angles and quantal adiabatic phase. J. Phys. A18, 15–27 (1985)Google Scholar
  7. Duistermaat, J.J.: On global action-angle coordinates. Commun. Pure Appl. Math.33, 687–706 (1980)Google Scholar
  8. Gerbert, P.: A systematic derivative expansion of the adiabatic phase. M.I.T. preprint., Center for Theoretical Physics (1988)Google Scholar
  9. Gotay, M., Lashof, R., Sniatycki, J., Weinstein, A.: Closed forms on symplectic fiber bundles. Commentarii Math. Helv.58, 617–621 (1983)Google Scholar
  10. Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press (1984)Google Scholar
  11. Hannay, J.H.: Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian. J. Phys. A18, 221–230 (1985)Google Scholar
  12. Jackiw, R.: Three elaborations on Berry's connection, curvature, and phase. Preprint (1987)Google Scholar
  13. Koiller, J.: Some remarks concerning Berry's phase. Seminario Brasileiro Analise (conference proceedings) SBA26 (1987a)Google Scholar
  14. Koiller, J.: The Foucault pendulum: an example of Berry's classical adiabatic angles. Preprint (1987b)Google Scholar
  15. Koiller, J.: Classical adiabatic angles for slowly moving mechanical systems. Preprint (1988)Google Scholar
  16. Kummer, M.: On the construction of the reduced phase space of a Hamiltonian system with symmetry. Indiana U. Math. J.30, 2, 281–291 (1981)Google Scholar
  17. Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys.5, 121–130 (1974)Google Scholar
  18. Marsden, J., Montgomery, R., Ratiu, T.: Hannay's angles for non-integrable and constrained systems. In progress (1988)Google Scholar
  19. Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry's phase. Phys. Rev. Lett.51, 2167 (1983)Google Scholar
  20. Takens, F.: Motion under the influence of a strong constraining potential. In: Global dynamical systems. Lecture Notes in Mathematics, vol. 819, p. 425–445. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  21. Vinet, Luc: Invariant Berry Connections. U. of Montréal preprint (1987)Google Scholar
  22. Weinstein, A.: Connections of Berry and Hannay type for moving Lagrangian submanifolds. Preprint (1988)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • R. Montgomery
    • 1
  1. 1.M.S.R.I.BerkeleyUSA

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