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