Symmetry, Reduction, and Phases in Mechanics 1

where (•) denotes the G-average. This principle is hard to justify in general and is probably only

justified for torus actions for integrable systems. We will only use it in this case in examples, so

we proceed with the form (5). Furthermore, we shall discard the term - (|| 2/j-12); we assume it is

small compared to the rest of the terms. Thus, define

Hq, P, 0 = \ lip

II2

- ^Zt) + V(q) + U(j(t)) = H0(q, p) - #(Zt) + U(tf(t)). (6)

The dynamics of Oi on the extended space T*Q x M is given by the vector field

(Xtf, Zt) = ^XHQ - X ^

Z t ) )

+ X(Uo mt), ZtJ. (7)

The vector field

hor(Zt) = (-X

(

^

Z t ) )

,Z

t

) (8)

has a natural interpretation as the horizontal lift of Zt relative to a connection, which we shall call

the Hannay-Berry connection induced by the Cartan connection; see §11 and §12,

especially Theorem 11.3. The holonomy of this connection is interpreted as the Hannay-Berry

phase of a slowly moving constrained system. Let us give a few more details for the case of the

ball in the rotating hoop.

§1B The ball in the rotating hoop

In the following example, we follow some ideas of J. Anandan.

Consider Figure 1B-1 which shows a hoop (not necessarily circular) on which a bead

slides without friction. As the bead is sliding, the hoop is slowly rotated in its plane through an

angle 0(t) and angular velocity co(t) = 8(t) k . Let s denote the arc length along the hoop,

measured from a reference point on the hoop and let q(s) be the vector from the origin to the

corresponding point on the hoop; thus the shape of the hoop is determined by this function q(s).

The unit tangent vector is q'(s) and the position of the reference point q(s(t)) relative to an

inertial frame in space is Re(t)q(s(t)), where Re is the rotation in the plane of the hoop through

an angle 9.