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Hamiltonian variational principles provided, since 60s, the means ofdeveloping very successful wave theories for nonlinear free-surface flows,under the assumption of irrotationality. This success, in conjunction with therecognition that almost all flows in the sea are not irrotational, raises thequestion of extending Hamilton Principle to rotational free-surface flows. Theequations governing the fluid motion within the fluid domain, in the Euleriandescription, have been derived by means of Hamilton Principle since late 50s.Nevertheless, a complete variational formulation of the problem, including thederivation of boundary conditions, seems to be lacking up to now. Such aformulation is given in the present work. The differential equations governingthe fluid motion are derived as usually, starting from the typical Lagrangian,constrained with the conservation of mass and the conservation of fluid parcelsidentity. To obtain the boundary conditions, generic differential-variationalconstraints are introduced in the boundary variational equation, leading to areformulation which permits us to derive both kinematic and dynamic conditionson all boundaries of the fluid, including the free surface. An interestingfeature, appearing in the present variational derivation of kinematic boundaryconditions, is a dual possibility of obtaining either the usual kinematiccondition (the same as in irrotational flow) or a condition of different type,corresponding to zero tangential velocity on the boundary. The deeper meaningand the significance of these findings seem to deserve further analysis.

Hamiltonian variational formulation of three-dimensional, rotational free-surface flows, with a moving seabed, in the Eulerian description