Symmetry, Reduction, and Phases in Mechanics 17
(The special case of this formula for a symmetric free rigid body was given by Hannay  and
Anandan , formula (20)).
horizontal lift of reduced
I f 1 r " ^
Figure 1G-1 For G =
(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)
A = e„
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
is given by