  We present a general theoretical framework for the exact treatment of a
hybrid system that is composed of a quantum subsystem and a classical
subsystem. When the quantum subsystem is dynamically fast and the classical
subsystem is slow, a vector potential is generated with a simple canonical
transformation. This vector potential, on one hand, gives rise to the familiar
Berry phase in the fast quantum dynamics; on the other hand, it yields a
Lorentz-like force in the slow classical dynamics. In this way, the pure phase
(Berry phase) of a wavefunction is linked to a physical force.
