1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

370 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY

A method that is more geometric in nature, is based on Montgomery's [1991b]

derivation of the phase formula for the rigid body and that for the polarization

phase shift. The general theory of this is found in Marsden, Montgomery and Ratiu

[1990]. Let MC C^3 be the manifold defined by setting the conserved quantities to

specific values (a level set of the momentum map). We will construct closed curves

in M in two pieces as follows.

The first portion c 0 is simply the dynamical trajectory in M joining two points

Po and P 1. It covers a closed curve in the reduced space. The invariants define the

reduction map W : C^3 ---+ JR^3 (the image is the three-wave surface) so that the

curve co projects onto a closed trajectory in the base space under W. (See Figure

2.13.)

: " dynamic trajectory

_,_....----r---

C1 __ __,,, M Co

Cz

phase P /\:

0 -

Figure 2.13. Curves used for the phase calculation.

In M we introduce three curves c 1 , c 2 , and c 3 that close the curve by

connecting the points P 1 and Po using the group actions: The curve c 1 goes

with the first S^1 action, etc. Then C1 u C2 closes the curve, as shown in the

figure, as do c1 U c3 and c2 U c3. Specifically, if Po = (q 1 ,q 2 ,q 3 ) and if P 1 =

(q1expi¢1,q2expi(¢1 +¢2),q3expi¢2), we let c 1 be the opposite (run in the op-

posite sense) of the curve

c~PP ( t) = ( q 1 exp it¢1, q2 exp it¢1, q3),

0 ::; t ::; 1, which is the action by t¢ 1 for the first group action and let c 2 be the

opposite of

c~PP(t) = (q 1 expi¢1, q2 expi(¢1 + t ¢2), q3 expit¢2)

which is the action by t¢2 for the second group action (These actions and the

corresponding conserved quantities were defined in the first lecture).

Let G be a canonical one form on C^3 , a scaling of the Poincare one form

defined by