LECTURE 1. REDUCTION THEORY 349The (cubic) Hamiltonian isHamilton's equations for a Hamiltonian H areand it is straightforward to check that Hamilton's equations are given in complex
notation by
dqk. 8H- = -2ZSk''(k-.
dt oi'fk
One checks that Hamilton's equations in our case coincide with the three wave
equations.
Integrals of motion. Besides H itself, there are additional constants of motion,
often referred to as the Manley-Rowe relations:
K1
lq112
+ lq2 l
2
S1 /1 S2/2
K2 lq2l2
+ lq 31
2
,
S2/2 S3/ 3
K 3
-----lq1l^2 lq31^2
S1 /1 S3/ 3The vector function ( K 1 , K 2 , K 3 ) is the momentum map for the following
symplectic action of the group T^3 = 81 x 81 x 81 on C^3 :
( q1, q2, q3 ) 1-7 ( q1 exp( i/1), q2 exp( i/1), q3),
( q1, q2, q3 ) 1-7 ( q1, q2 exp( i/2), q3 exp( i/2)),
( q1, q2, q3 ) 1-7 ( q1 exp( if3), q2, q3 exp( -i/3) ).The Hamiltonian taken with any two of the K 1 are checked to be a complete
and independent set of conserved quantities. Thus, the system is Liouville-Arnold
integrable.
The K 1 clearly give only two independent invariants since K1 - K2 = K 3. Any
combination of two of these actions can be generated by the third reflecting the fact
that the K 1 are linearly dependent. Another way of saying this is that the group
action by T^3 is really captured by the action of T^2.
Integrating the equations. To carry out the integration, one can make use of the
Hamiltonian plus two of the integrals, K 1 to reduce the system to quadratures. This
is often carried out using the transformation q 1 = ..Ji5j exp i</> 1 to obtain expressions
for the phases ¢ 1. The resulting expressions are nice, but the alternative point of
view using invariants is also useful.
Poisson reduction. Symplectic reduction of the above Hamiltonian system uses
the symmetries and associated conserved quantities K k. In Poisson reduction, we