Symmetry, Reduction, and Phases in Mechanics 17
(The special case of this formula for a symmetric free rigid body was given by Hannay [1985] and
Anandan [1988], formula (20)).
dynamic phase
holonomy
(geometric phase)
true (reconstructed)
trajectory
InJ"1^)
horizontal lift of reduced
trajectory
reduced trajectory
I f 1 r " ^
Figure 1G-1 For G =
S1,
(log holonomy) = TT-TT © , (log dynamic phase) = - £(t) dt, where
^ IIM-II ^D ^ IIM-II Jo
T = period of reduced trajectory and co = reduced symplectic form.
To prove (2), we choose the connection one-form on J *(u.) to be (see Proposition 2.2)
1
A = e„
(3)
where 0 is the pull back to J_1(M.) of the canonical one-form 0 on T*SO(3). The curvature of
A as a two-form on the base P„, the sphere of radius || \i \\ in
R3,
is given by
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