Hamiltonian formulation of nonlinear water waves in a two-fluid system

In this paper, it is dealt with that the Hamilton formulation of nonlinear water waves in a two-fluid system, which consists of two layers of constant -density incompressible inviscid fluid with a horizontal bottom, an interfa ce and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid -layer and the reduced kinetic thickness of upper fluid-layer as the genera lized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legend re transformation under the shallow water assumption. Hence the results for single-layer fluid are extended to the case of stratified fluid.