Hamiltonian formulations for surface waves

Abstract

The formulation of Hamilton's principle for the Eulerian description of the motion of an ideal fluid leads to the identification of the pressure as a Lagrangian density. The resulting variational principle for irrotational gravity waves on the surface of a homogeneous fluid yields both Laplace's equation in the interior and the free-surface boundary conditions. The velocity potential at, and the displacement of, the free surface are canonical variables in Hamilton's sense. The corresponding canonical equations are of Boussinesq's type in the regime of weak dispersion and weak nonlinearity; they are valid only for relatively long waves but may be either stable or unstable with respect to short waves, depending on whether the corresponding Hamiltonian is or is not positive definite, as originally pointed out by Broer. The Korteweg-deVries and related equations are discussed.