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].