On variational formulation in water wave mechanics

In this paper variational formulations for surface gravitational waves in inviscid incompressible fluids are investigated. The formulations are introduced with the help of the principle of virtual work. The starting point are equations of motion multiplied by a field of virtual displacements and integrated over the region occupied by the fluid. In derivations of the virtual work equation careful attention is paid to mutual relations between Eulerian and Lagrangian descriptions. The integration of the equation with respect to time leads to the expression for the Lagrangian function and then the Hamilton’s principle. The case of a potential flow and spatial description provides a generalisation of the Lagrangian given by Luke (1967). Variational formulations of equations of fluid dynamics are especially important in developing approximate discrete descriptions of an original task. The formulations enable us to construct numerical models preserving some important features of the original system. With the variational formulations equations of the discrete models can be obtained by direct variation of a set of discrete parameters in the action integral associated with the problem considered. In the literature on the subject, variational formulations for the dynamics of perfect fluids have been presented in several papers. Herivel (1955) gave Hamilton’s principle in two variational formulations corresponding to the Lagrangian and Eulerian variables, respectively. The same problem of formulation of Hamilton’s principle for perfect fluids is discussed in Serrin’s monograph (1959). Serrin has found that the formulation due to Herivel is satisfactory only for the Lagrangian variables. As concerns the Eulerian description, the Herivel solution should be supplemented with an additional constraint of conservation of the fluid particles identities. The correct version of the variational principle obtained in this way, called the Herivel-Lin principle, takes into account the conservation of