Various holonomy phenomena are shown to be instances of the reconstruction procedure
for mechanical systems with symmetry. We systematically exploit this point of view for fixed
systems (for example with controls on the internal, or reduced, variables) and for slowly moving
systems in an adiabatic context. For the latter, we obtain the phases as the holonomy for a
connection which synthesizes the Cartan connection for moving mechanical systems with the
Hannay-Berry connection for integrable systems. This synthesis allows one to treat in a natural
way examples like the ball in the slowly rotating hoop and also non-integrable mechanical systems.
Acknowledgements We thank J. Anandan, A. Fischer, J. Koiller, P. Krishnaprasad, R.
Littlejohn, A. Pines, A. Weinstein, and the referee for their valuable comments.
Manuscript Received April 10, 1989 In Revised Form: January 5, 1990
J.E. Marsden, Department of Mathematics, University of California, Berkeley, CA 94720.
Research partially supported by NSF grant DMS 8702502 and DOE Contract DE-AT03-88ER-
R. Montgomery, Department of Mathematics, University of California, Santa Cruz, CA 95064.
Research partially supported by an NSF postdoctoral fellowship and NSF grant DMS 8702502.
T. Ratiu, Department of Mathematics, University of California, Santa Cruz, CA 95064.
Research partially supported by NSF grant DMS 8701318-01, and AFOSR/DARPA contract