Classical adiabatic holonomy in a Grassmannian system

The ‘‘geometrical phase factor,’’ found by Berry in his study of the quantum adiabatic theorem, has a classical analogue discovered recently by Hannay and Berry himself. This classical analogue consists of an extra angle swept by the system in its classical evolution during an adiabatic excursion. As in Berry’s phase, Hannay’s angle reflects a nontrivial holonomy of the system and it has a purely geometric character. In this paper we calculate Hannay’s angles for the classical Grassmannian system associated to a quantum spin in a magnetic field. We also show their close relation to the corresponding quantum-mechanical phase of Berry for the same system.