Symmetry, Reduction, and Phases in Mechanics 15 imaginary part of the inner product) via reduction, as in Abraham and Marsden [1978]. We shall show in §4 that this formula is the holonomy of the closed loop relative to a principal S1- connection on complex Hilbert space and is a particular case of the holonomy formula in principal bundles with abelian structure group. Littlejohn [1988] has shown that the Bohr-Sommerfeld and Maslov phases of semi- classical mechanics can be viewed as incarnations of Berry's phase. To do this he notes that Gaussian wave-packets define an embedding of classical phase space into Hilbert space, then uses the Aharonov-Anandan point of view on phases, together with the variational formulation of quantum mechanics. The quantum-classical relation between the phases is also considered in Hannay [1985], Anandan [1988], and Weinstein [1989a,b]. §1F Integrable systems Consider an integrable system with action-angle variables (I p I2, ..., In, Qv 82, — » ®n) and with a Hamiltonian H(Ij, I2, ... In, Bv 82,... 9n m) that depends on a parameter m e M. Let c be a loop based at a point m0 in M. We want to compare the angular variables in the torus over m0, once the system is slowly changed as the parameters undergo the circuit c. Since the dynamics in the fiber varies as we move along c, even if the actions vary by a negligible amount, there will be a shift in the angle variables due to the frequencies co1 = 3H/3I1 of the integrable system correspondingly, one defines dynamic phase = co^I, c(t))) dt . Jo Here we assume that the loop is contained in a neighborhood whose standard action coordinates are defined. In completing the circuit c , we return to the same torus, so a comparison between the angles makes sense. The actual shift in the angular variables during the circuit is the dynamic phase plus a correction term called the geometric phase. One of the key results is that this geometric phase is the holonomy of an appropriately constructed connection called the Hannay- Berry connection on the torus bundle over M which is constructed from the action-angle variables. The corresponding angular shift, computed by Hannay [1985], is called Hannay's angles, so the actual phase shift is given by I A8 = dynamic phases + Hannay's angles . The geometric construction of the Hannay-Berry connection for classical systems is given in terms of momentum maps and averaging in Golin, Knauf, and Marmi [1989] and Montgomery [1988].

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