  A Hamiltonian reduction approach is defined, studied, and finally used to
derive asymptotic models of internal wave propagation in density stratified
fluids in two-dimensional domains. Beginning with the general Hamiltonian
formalism of Benjamin [1] for an ideal, stably stratified Euler fluid, the
corresponding structure is systematically reduced to the setup of two
homogeneous fluids under gravity, separated by an interface and confined
between two infinite horizontal plates. A long-wave, small-amplitude
asymptotics is then used to obtain a simplified model that encapsulates most of
the known properties of the dynamics of such systems, such as bidirectional
wave propagation and maximal amplitude travelling waves in the form of fronts.
Further reductions, and in particular devising an asymptotic extension of
Dirac's theory of Hamiltonian constraints, lead to the completely integrable
evolution equations previously considered in the literature for limiting forms
of the dynamics of stratified fluids. To assess the performance of the
asymptotic models, special solutions are studied and compared with those of the
parent equations.
