Cartan-Hannay-Berry Phases and Symmetry
Marsden, J. E, R. Montgomery, and T. Ratiu
Contemporary Mathematics, AMS, 97, (1989), 279-295
We give a systematic treatment of the treatment of the classical Hannay-Berry phases for mechanical systems in terms of the holonomy of naturally constructed connections on bundles associated to the system. We make the costructions using symmetry and reduction and, for moving systems, we use the Cartan connection. These ideas are woven with the idea of Montgomery  on the averaging of connections to produce the Hannay-Berry connection.