The analysis provided by Berry of the extra phase acquired by a system undergoing adiabatic excursion in a closed path in parameter space is extended in this paper to the case of nonstationary states. It is shown that, for certain integrable dynamical systems, wave packets exist which under continuation in parameter space behave in a manner that can be understood entirely in classical terms, thus providing a connection between the quantum adiabatic phase and the classical Hannay angle shifts.