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.
Similar content being viewed by others
References
Anandan, J.: Geometric angles in quantum and classical physics. Phys. Lett. A (in press 1988)
Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull Lond. Math. Soc.14, 1–15 (1982)
Arnold, V.I.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1978
Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations. Berlin, Heidelberg, New York: Springer 1983
Berry, M.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A392, 45 (1984)
Berry, M.: Classical adiabatic angles and quantal adiabatic phase. J. Phys. A18, 15–27 (1985)
Duistermaat, J.J.: On global action-angle coordinates. Commun. Pure Appl. Math.33, 687–706 (1980)
Gerbert, P.: A systematic derivative expansion of the adiabatic phase. M.I.T. preprint., Center for Theoretical Physics (1988)
Gotay, M., Lashof, R., Sniatycki, J., Weinstein, A.: Closed forms on symplectic fiber bundles. Commentarii Math. Helv.58, 617–621 (1983)
Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press (1984)
Hannay, J.H.: Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian. J. Phys. A18, 221–230 (1985)
Jackiw, R.: Three elaborations on Berry's connection, curvature, and phase. Preprint (1987)
Koiller, J.: Some remarks concerning Berry's phase. Seminario Brasileiro Analise (conference proceedings) SBA26 (1987a)
Koiller, J.: The Foucault pendulum: an example of Berry's classical adiabatic angles. Preprint (1987b)
Koiller, J.: Classical adiabatic angles for slowly moving mechanical systems. Preprint (1988)
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)
Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys.5, 121–130 (1974)
Marsden, J., Montgomery, R., Ratiu, T.: Hannay's angles for non-integrable and constrained systems. In progress (1988)
Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry's phase. Phys. Rev. Lett.51, 2167 (1983)
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 1979
Vinet, Luc: Invariant Berry Connections. U. of Montréal preprint (1987)
Weinstein, A.: Connections of Berry and Hannay type for moving Lagrangian submanifolds. Preprint (1988)
Author information
Authors and Affiliations
Additional information
Communicated by B. Simon
Rights and permissions
About this article
Cite this article
Montgomery, R. The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case. Commun.Math. Phys. 120, 269–294 (1988). https://doi.org/10.1007/BF01217966
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01217966