We present a critical review of the Hamiltonian and the Lagrangian
theories of
pattern formation in driven capillary waves at low viscosity and high aspect
ratio.
We construct a Hamiltonian perturbation theory in the spirit of Milner's
(1991)
formulation, and derive the amplitude equations and their coefficients
relevant at
the onset of surface waves. Our presentation is detailed, and we carefully
point
out the differences between our results for the nonlinear
coefficients and the results
obtained by others. From our standing wave analysis we find that the square
pattern
is subcritical. Among the supercritical standing wave patterns, we find
that the
eightfold quasi-crystalline pattern, observed by Christiansen
et al. (1992) and by
Bosch (1995), is more stable than both rolls and hexagons. We outline the
high-aspect-ratio experimental results obtained so far, and discuss them
in the
light of the theory.