  A method for extracting finite-dimensional Hamiltonian systems from a class
of 2+1 Hamiltonian mean field theories is presented. These theories possess
noncanonical Poisson brackets, which normally resist Hamiltonian truncation,
but a process of beatification by coordinate transformation near a reference
state is described in order to perturbatively overcome this difficulty. Two
examples of four-wave truncation of Euler's equation for scalar vortex dynamics
are given and compared: one a direct non-Hamiltonian truncation of the
equations of motion, the other obtained by beatifying the Poisson bracket and
then truncating.
