Some examples of a new kind of the Aharonov-Anandan phase which is of mechanical origin are studied. It is the non-adiabatic generalization of the mechanical Berry phase. We first study the periodic motion and then the interference effect. Finally we discuss the relation between Aharonov-Anandan phase and Berry phase.
Berry's (1984) formulation of the topological phase for periodic Hamiltonians is extended to the case of non-adiabatic evolution by proper choice of initial states. This restores a full parallel with the non-adiabatic formalism of Aharonov and Anandan (1987) for periodic orbits in a projective Hilbert space. The Berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by Aharonov and Anandan and by Page (1987).