18 Marsden, Montgomery, and Ratiu

- — c o = — ^ d S . (4)

mi * nmi2

The first terms in (2) represent the geometric phase, i.e., the holonomy of the reduced trajectory

with respect to this connection. By Corollary 4.2, the logarithm of the holonomy (modulo 2n) is

given as minus the integral over D of the curvature, i.e., it equals

- L f co,, = - — ^ r (areaD) = -A(mod27i) (5)

I W | J D * HUH2

The second terms in (2) represent the dynamic phase. By the algorithm of Proposition 2.1

it is calculated in the following way. First one horizontally lifts the reduced closed trajectory II(t)

to

J-1(|i)

relative to the connection (3). This horizontal lift is easily seen to be (identity, Il(t)) in

the left trivalization of T*SO(3) as SO(3) x

R3.

Second, one computes

§(t) = (A-XH)(n(t)). (6)

Since in coordinates

en = X PiW a n d x H = I p i 7 T + r terms

i i

oq1

dp

for p* = Xs^Pj' $ being the inverse of the Riemannian metric g- on SO(3), we get

J

(9R-XH)(n(t)) = I i P i = 2H(identity, n(t)) = 2H^( (7)

i

where H is the value of the energy on S2 along the integral curve I~I(t). Consequently,

2H

Third, since ^(t) is independent of t, the solution of the equation

2H

g

=

& = ZT g£

is

8(t) = exp

so that the dynamic phase equals

^2Hut ^

—^ C

vlWI j

2H

A0d = —^ T (mod

2TC)

(9)

INI

Formulas (5) and (9) prove (2). Note that (2) is independent of which spherical cap one chooses

amongst the two bounded by II(t). Indeed, the solid angles on the unit sphere defined by the two

caps add to 4n, which does not change formula (2).