Introduction This paper is concerned with the interpretation of the Hannay-Berry phase for classical mechanical systems as the holonomy of a connection on a bundle associated with the given problem. The techniques apply to the quantum case in the spirit of Aharonov and Anandan [1987], Anandan [1988] and Simon [1983] using the well known fact that quantum mechanics can be regarded as an instance of classical mechanics (see for instance Abraham and Marsden [1978]). In carrying this out there are a number of interesting new issues beyond that found in Hannay [1985] and Berry [1984], [1985] that arise. Already this is evident for the example of the ball in the hoop discussed in Berry [1985] some remarks on this example are discussed in §1 below. For slowly varying integrable systems and for some aspects of the nonintegrable case, progress was made already by Golin, Knauf, and Marmi [1989] and Montgomery [1988]. The situation for the integrable case has been generalized to the context of families of Lagrangian manifolds by Weinstein [1989a,b]. However, these do not satisfactorily cover even the ball in the hoop example. For this and other examples, there is need for a development of the formulation, and it is the purpose of this paper to give one, following the line of investigation initiated by these papers. One of the crucial new ingredients in the present paper is the introduction of a connection that is associated to the movement of a classical system that we term the Cartan connection. It is related to the theory of classical spacetimes that was developed by Cartan [1923] (see for example, Marsden and Hughes [1983] for an account). Another ingredient is the systematic use of symmetry and reduction, which are the key concepts needed to generalize to the nonintegrable case. In fact it is through the reconstruction process that the holonomy enters. The paper begins in §1 with some simple examples. The purpose is to give an idea of the Cartan connection. The first example is the ball in the hoop. The second example is the problem of two coupled rigid bodies to illustrate some of the ideas involved in reconstruction (here there are no slowly varying parameters, but there is still holonomy). We also give the Aharonov-Anandan formula for quantum mechanics (given in detail in §4) and a resume of slowly varying integrable systems from Golin, Knauf, and Marmi [1989] and Montgomery [1988]. Finally, we give the example of reconstructing the motion of a freely spinning rigid body. §§2 and 3 deal with the general theory of reconstruction. Given a phase space P and a symmetry group G, we show how to reconstruct the dynamics on P from dynamics on the reduced spaces. If J : G Q* is an equivariant momentum map for the G-action, the reduced space is P = J~l(\x)/G , where G is the coadjoint isotropy at \i. This reconstruction is done using a choice of connection on the principal G -bundle J_1(M-) » ?M (assuming the action is free). In case P is a cotangent bundle, there is a family of natural choices of connections 1
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