The integrable structure of the three-wave equations is discussed in the setting
of geometric mechanics. Lie-Poisson structures with quadratic Hamiltonian
are associated with the three-wave equations through the Lie algebras
and
. A second structure having cubic Hamiltonian is shown to be compatible.
The analogy between this system and the rigid-body or Euler equations is
discussed. Poisson reduction is performed using the method of invariants and
geometric phases associated with the reconstruction are calculated. We show
that using piecewise continuous controls, the transfer of energy among three
waves can be controlled. The so called quasi-phase-matching control strategy,
which is used in a host of nonlinear optical devices to convert laser light from
one frequency to another, is described in this context. Finally, we discuss the
connection between piecewise constant controls and billiards.
