10

Marsden, Montgomery, and Ratiu

4TCA which means that on average, the shift in position is by —r~" between the rotated and nonrotated

hoop. Note that if co0 = 0 (the situation assumed by Berry [1985]) then averaging over initial

conditions is not necessary. This process of averaging over the initial conditions that we naturally

encounter in this example is related to the recent work of Golin and Marmi [1989] on experimental

procedures to measure the phase shift.

This extra length -^— is sometimes called the Hannay-Berry phase. Expressed in

O L A

angular measure, it is —^- • m §HB we show, using the Cartan connection, how to realize this

answer as the holonomy of the associated Hannay-Berry connection.

§1C Coupled planar pendula

We return now to an example similar to Elroy's beanie, with which we began. Consider

two coupled pendula in the plane moving under the influence of a potential depending on the hinge

angle between them. Let rx, r2 be the distances from the joint to their centers of mass and let Qx

and 02 be the angles formed by the straight lines through the joint and their centers of mass

relative to an inertial coordinate system fixed in space, as in Figure 1C-1. The Lagrangian of this

system is

L = \ m^G 2 + | m ^ S 2 - V ^ - 82)

and is therefore of the form kinetic minus potential energy, where the kinetic energy is given by the

metric on

R2

ds2

= m ^ d9f + rry! d0^ .

Figure 1C-1