The formalism of classical analytical dynamics is used in conjunction with the principle of virtual velocity to derive Lagrangian expressions for the hydrodynamic forces acting on a rigid body moving through an in-viscid and incompressible liquid with a free surface. Simultaneously, a corresponding Lagrangian expression is derived for the hydrodynamic pressure acting on the free surface itself. The expressions for the hydrodynamic forces degenerate to the classical ones if the free surface is not present, and the expression for the pressure is reduced to that obtained by Milder if the rigid boundaries are all kept fixed. The derived Lagrangian expressions for the hydrodynamic reactions are used to obtain a complete set of motion equations for the examined hydromechanical system, and to discuss another Lagrangian approach to the ship-motions problem, presented by Wang.
Lagrangian Expressions of the Hydrodynamic Forces Acting on a Rigid Body in the Presence of a Free Surface
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Athanassoulis, G. A., and T. A. Loukakis. "Lagrangian Expressions of the Hydrodynamic Forces Acting on a Rigid Body in the Presence of a Free Surface." J Ship Res 29 (1985): 12–22. doi: https://doi.org/10.5957/jsr.1922.214.171.124
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