2 Marsden, Montgomery, and Ratiu depending on a choice of metric on the configuration space and on a choice of transverse cross section to the G -orbit. We shall later refer to this one as the mechanical connection. Another one is built out of the canonical one-form and applies when G is abelian. The mechanical connection connection was defined by Guichardet [1984] and is closely related to connections defined by Smale [1970] and Kummer [1981]. For the case of cotangent bundles of semisimple Lie groups, the first includes the second as a special case. We treat both the Lagrangian and Hamiltonian cases, since in the former the procedure is considerably more concrete because the Euler-Lagrange equations are of second order. In these sections there are no slowly varying parameters, but there are connections and holonomy. The connections are combined with the Hannay-Berry construction in §14. §4 gives,for the convenience of the reader, background material on Ehresmann connections, curvature and holonomy that is needed for the paper. It is illustrated with the Aharonov-Anandan formula and other examples of mechanical systems—such as the top in a gravitational field and coupled planar rigid bodies in §5. In §§6, 7 and 8 we present the basic defining properties and the existence and uniqueness of the Hannay-Berry connection. A crucial aspect of the construction is to take a given connection and average it relative to the action of a group G. This action also defines a parametrized momentum map I, which plays an important role. We generalize the theory developed in Montgomery [1988], and Golin, Knauf, and Marmi [1989] for trivial bundles with symplectic fibers and the standard connection to nontrivial fiber bundles whose fibers are Poisson manifolds and with a connection compatible with this structure. It is important to allow a nontrivial connection at this stage, even if the bundle is trivial, in order to deal with moving systems, like the ball in the hoop. For moving systems, the nontrivial connection used is the Cartan connection described in §11. §9 gives another way to look at the Hannay-Berry connection by utilizing the momentum map for the group G. §10 studies the important case of slowly moving integrable systems. This is the case that motivated the development in Montgomery [1988], Golin, Knauf, and Marmi [1989], and Weinstein [1989a,b]. We generalize this to our context. §12 presents a general construction for inducing connections on a tower of two bundles: E — F — » M with a given a connection on E — M and a family of fiberwise connections on E — F. This is applied in §13 with E = I_1(|i), F = I_1(|i)/G , and M the parameter space. This is a parametrized version of the bundle of reduced spaces. This construction allows us to glue together the Hannay-Berry connection and the connection on the bundle J-1(M) — » P„ used in §§2 and 3 to obtain a connection on I-1(M) — I-1(M-)/G . The holonomy of this synthesized connection gives the desired phase changes in many of the equations. In this paper, there are three lines of investigation one can focus on if desired. We regard §4 on Ehresmann connections as necessary background for all three. The three lines are: 1 Reconstruction ideas: §1C, D, F,§2, 3, 5,13 2 Adiabatic phases and moving systems: §1A, B, E, 6, 7, 8, 9, 10, 11, 12 3 Synthesis and future directions: §13, 14.

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