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
Previous Page Next Page