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  and is closely related to connections
defined by Smale  and Kummer . 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 , and Golin, Knauf, and Marmi
 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 , Golin, Knauf, and Marmi , 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
— » 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.