350 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
obtain the symplectic leaves in this reduction, we use the method of invariants.
Invariants for the T^2 action are:
X + iY = q1i'f2q3
Z1 = lq1l
2- lq2l
2
Z2 = lq2l^2 - lq 312 ·These quantities provide coordinates for the four dimensional orbit space C^3 /T^2.
The following identity (this is part of the invariant theory game) holds for these
invariants and the conserved quantities:
where the constants (3, 8 are given by
(3 _ S1/1S2/2S3/3 8 = s212K1 + s313(K1 - K2)·
- (s212 + S3/3)^3 '
This defines a two dimensional surface in (X, Y, Z2) space, with Z1 determined
by the values of these invariants and the conserved quantities (so it may also be
thought of as a surface in (X, Y, Z 1 , Z 2 ) as well). A sample of one of these surfaces
is plotted in Figure 1.2.
yx
Figure 1.2. The reduced phase space for the three-wave equations.We call these surfaces the three wave surfaces. They are examples of orb-
ifolds. The evident singularity in the space is typical of orbifolds and comes about
from the non-freeness of the group action.
Any trajectory of the original equations defines a curve on each three wave sur-
face, in which the Kj are set to constants. These three wave surfaces are the sym-
plectic leaves in the four dimensional Poisson space with coordinates ( X, Y, Z 1 , Z 2 ).