LECTURE 1. REDUCTION THEORY 351The original equations define a dynamical system in the Poisson reduced space
and on the symplectic leaves as well. The reduced Hamiltonian is
H(X,Y,Z 1 ,Z2) = -X
and indeed, X = 0 is one of the reduced equations. Thus, the trajectories on the
reduced surfaces are obtained by slicing the surface with the planes X = Constant.
The Poisson structure on C^3 drops to a Poisson structure on (X, Y, Z 1 , Z 2 )-space,
and the symplectic structure drops to one on each three wave surface-this is
of course an example of the general procedure of symplectic reduction. Also,
fro'm the geometry, it is clear that interesting homoclinic orbits pass through the
singular points-these are cut out by the plane X = 0.
A control perspective allows one to manipulate the plane H = -X and
thereby the dynamics. This aspect is explored in Alber, Luther, Marsden and
Robbins [1998a].