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
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).
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