2 A variational principle for the interior fluid motion

The configuration of the fluid in a rotating–translating vessel is shown in figure 1. The fluid occupies the region $0\leqslant y\leqslant \unicode[STIX]{x1D702}(x,t)$ , with $0\leqslant x\leqslant L$ . The two-dimensional Euler equations relative to a rotating–translating coordinate system $(x,y)$ given by Alemi Ardakani & Bridges (Reference Alemi Ardakani and Bridges2012) are

(2.1) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\text{D}u}{\text{D}t}+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}=\displaystyle -g\sin \unicode[STIX]{x1D703}+2\dot{\unicode[STIX]{x1D703}}v+\ddot{\unicode[STIX]{x1D703}}(y+d_{2})+\dot{\unicode[STIX]{x1D703}}^{2}(x+d_{1})-\ddot{q}_{1}\cos \unicode[STIX]{x1D703}-\ddot{q}_{2}\sin \unicode[STIX]{x1D703},\\ \displaystyle \frac{\text{D}v}{\text{D}t}+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}=\displaystyle -g\cos \unicode[STIX]{x1D703}-2\dot{\unicode[STIX]{x1D703}}u-\ddot{\unicode[STIX]{x1D703}}(x+d_{1})+\dot{\unicode[STIX]{x1D703}}^{2}(y+d_{2})+\ddot{q}_{1}\sin \unicode[STIX]{x1D703}-\ddot{q}_{2}\cos \unicode[STIX]{x1D703},\\ \displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y}=0,\end{array}\right\}\end{eqnarray}$$

where $\text{D}u/\text{D}t=u_{t}+uu_{x}+vu_{y}$ . The velocity field $(u,v)$ is relative to the body frame and $p$ is the pressure field. Relative to the body frame, the boundary conditions are

(2.2a-c ) $$\begin{eqnarray}u=0\quad \text{at }x=0\quad \text{and}\quad x=L,\quad v=0\quad \text{at }y=0\end{eqnarray}$$

and

(2.3a,b ) $$\begin{eqnarray}p=0\quad \text{and}\quad \unicode[STIX]{x1D702}_{t}+u\unicode[STIX]{x1D702}_{x}=v\quad \text{at }y=\unicode[STIX]{x1D702}(x,t),\end{eqnarray}$$

where the surface tension is neglected in the boundary condition for the pressure. The vorticity, ${\mathcal{V}}=v_{x}-u_{y}$ , satisfies the equation $\text{D}{\mathcal{V}}/\text{D}t=-2\ddot{\unicode[STIX]{x1D703}}$ .

Now, we introduce a velocity potential $\unicode[STIX]{x1D719}(x,y,t)$ such that

(2.4a,b ) $$\begin{eqnarray}u(x,y,t)=\unicode[STIX]{x1D719}_{x}+\dot{\unicode[STIX]{x1D703}}(y+d_{2})\quad \text{and}\quad v(x,y,t)=\unicode[STIX]{x1D719}_{y}-\dot{\unicode[STIX]{x1D703}}(x+d_{1}).\end{eqnarray}$$

The velocity field in (2.4) satisfies the vorticity equation. The vorticity is constant in space and satisfies ${\mathcal{V}}=-2\dot{\unicode[STIX]{x1D703}}$ . Substitution of (2.4) into the continuity equation in (2.1) leads to Laplace’s equation for $\unicode[STIX]{x1D719}(x,y,t)$ ,

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{xx}+\unicode[STIX]{x1D719}_{yy}=0\quad \text{in }0\leqslant y\leqslant \unicode[STIX]{x1D702}(x,t),0\leqslant x\leqslant L,\end{eqnarray}$$

and substitution of (2.4) into the momentum equations in (2.1) leads to Bernoulli’s equation for the pressure field,

(2.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \frac{p}{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D719}_{t}+\frac{1}{2}(\unicode[STIX]{x1D719}_{x}^{2}+\unicode[STIX]{x1D719}_{y}^{2})-\frac{1}{2}(x+d_{1})(-g\sin \unicode[STIX]{x1D703}-\ddot{q}_{1}\cos \unicode[STIX]{x1D703}-\ddot{q}_{2}\sin \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle -\,\frac{1}{2}(y+d_{2})(-g\cos \unicode[STIX]{x1D703}+\ddot{q}_{1}\sin \unicode[STIX]{x1D703}-\ddot{q}_{2}\cos \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \quad +\,\dot{\unicode[STIX]{x1D703}}(y+d_{2})\unicode[STIX]{x1D719}_{x}-\dot{\unicode[STIX]{x1D703}}(x+d_{1})\unicode[STIX]{x1D719}_{y}=Be(t),\end{eqnarray}$$

where $Be(t)$ is the Bernoulli function which can be absorbed into $\unicode[STIX]{x1D719}(x,y,t)$ . Therefore, the dynamic free-surface boundary condition in (2.3) at $y=\unicode[STIX]{x1D702}(x,t)$ becomes

(2.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D719}_{t}+{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D719}_{x}^{2}+\unicode[STIX]{x1D719}_{y}^{2})+\dot{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D702}+d_{2})\unicode[STIX]{x1D719}_{x}-{\textstyle \frac{1}{2}}(x+d_{1})(-g\sin \unicode[STIX]{x1D703}-\ddot{q}_{1}\cos \unicode[STIX]{x1D703}-\ddot{q}_{2}\sin \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D702}+d_{2})(-g\cos \unicode[STIX]{x1D703}+\ddot{q}_{1}\sin \unicode[STIX]{x1D703}-\ddot{q}_{2}\cos \unicode[STIX]{x1D703})-\dot{\unicode[STIX]{x1D703}}(x+d_{1})\unicode[STIX]{x1D719}_{y}=0.\end{eqnarray}$$

In terms of the velocity potential $\unicode[STIX]{x1D719}(x,y,t)$ , the kinematic free-surface boundary condition in (2.3) becomes

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{t}+(\unicode[STIX]{x1D719}_{x}+\dot{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D702}+d_{2}))\unicode[STIX]{x1D702}_{x}=\unicode[STIX]{x1D719}_{y}-\dot{\unicode[STIX]{x1D703}}(x+d_{1})\quad \text{at }y=\unicode[STIX]{x1D702}(x,t)\end{eqnarray}$$

and the rigid-wall boundary conditions in (2.2) become

(2.9a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{x}=-\dot{\unicode[STIX]{x1D703}}(y+d_{2})\quad \text{at }x=0,L,\quad \unicode[STIX]{x1D719}_{y}=\dot{\unicode[STIX]{x1D703}}(x+d_{1})\quad \text{at }y=0.\end{eqnarray}$$

Following Luke, the variational principle for the interior rotating–translating fluid is

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}{\mathcal{L}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D702})=\unicode[STIX]{x1D6FF}\int _{t_{1}}^{t_{2}}\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}(x,t)}p(x,y,t)\,\text{d}y\,\text{d}x\,\text{d}t=0.\end{eqnarray}$$

Now, substituting for $p(x,y,t)$ from (2.6) into (2.10), we obtain (1.8). Taking the variations in (1.8), we obtain

(2.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D6FF}{\mathcal{L}}(\unicode[STIX]{x1D719},\unicode[STIX]{x1D702})=\displaystyle \int _{t_{1}}^{t_{2}}\int _{0}^{L}(\unicode[STIX]{x1D702}_{t}+\unicode[STIX]{x1D702}_{x}\unicode[STIX]{x1D719}_{x}-\unicode[STIX]{x1D719}_{y}+\unicode[STIX]{x1D702}_{x}\dot{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D702}+d_{2})+\dot{\unicode[STIX]{x1D703}}(x+d_{1}))\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}\big|^{y=\unicode[STIX]{x1D702}}\unicode[STIX]{x1D70C}\,\text{d}x\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle +\,\displaystyle \int _{t_{1}}^{t_{2}}\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}(x,t)}(\unicode[STIX]{x1D719}_{xx}+\unicode[STIX]{x1D719}_{yy})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x\,\text{d}t-\int _{t_{1}}^{t_{2}}\int _{0}^{L}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\boldsymbol{\cdot }\boldsymbol{n}+\dot{\unicode[STIX]{x1D703}}(x+d_{1}))\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}\big|_{y=0}\unicode[STIX]{x1D70C}\,\text{d}x\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle +\,\int _{t_{1}}^{t_{2}}\int _{0}^{\unicode[STIX]{x1D702}}(-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\boldsymbol{\cdot }\boldsymbol{n}+\dot{\unicode[STIX]{x1D703}}(y+d_{2}))\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}\big|_{x=0}\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle +\,\int _{t_{1}}^{t_{2}}\!\int _{0}^{\unicode[STIX]{x1D702}}(-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}\boldsymbol{\cdot }\boldsymbol{n}-\dot{\unicode[STIX]{x1D703}}(y+d_{2}))\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}\big|_{x=L}\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}t+\!\int _{t_{1}}^{t_{2}}\!\int _{0}^{L}p(x,\unicode[STIX]{x1D702},t)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}\,\text{d}x\,\text{d}t=0.\end{eqnarray}$$

A detailed derivation of (2.11) is given in appendix A. From (2.11), it is obvious that invariance of ${\mathcal{L}}$ with respect to a variation in the free-surface elevation $\unicode[STIX]{x1D702}$ yields the dynamic free-surface boundary condition (2.7). Similarly, the invariance of ${\mathcal{L}}$ with respect to a variation in the velocity potential $\unicode[STIX]{x1D719}$ yields the field equation (2.5). Moreover, the invariance of ${\mathcal{L}}$ with respect to a variation in the velocity potential $\unicode[STIX]{x1D719}$ at $y=0$ , $x=0$ and $x=L$ recovers the rigid-wall boundary conditions in (2.9), and the invariance of ${\mathcal{L}}$ with respect to a variation in the velocity potential $\unicode[STIX]{x1D719}$ at $y=\unicode[STIX]{x1D702}$ recovers the kinematic free-surface boundary condition (2.8).

3 A variational principle for the exterior water waves and the motion of the rigid body containing fluid

The complete set of differential equations for the exterior water waves interacting with the rigid body in the plane can be obtained from a variant of the Lagrangian action (1.3) which takes the form

(3.1) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D6FF}{\mathcal{L}} & = & \displaystyle \displaystyle \unicode[STIX]{x1D6FF}\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}(t)}-\unicode[STIX]{x1D70C}\big(\unicode[STIX]{x1D6F7}_{t}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}+gY\big)\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle +\,\unicode[STIX]{x1D6FF}\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}^{\prime }}(\mathsf{KE}^{fluid}-\mathsf{PE}^{fluid})\,\text{d}\unicode[STIX]{x1D6FA}^{\prime }\,\text{d}t+\unicode[STIX]{x1D6FF}\int _{t_{1}}^{t_{2}}(\mathsf{KE}^{vessel}-\mathsf{PE}^{vessel})\,\text{d}t=0,\end{eqnarray}$$

where the second line in (3.1) is a 2D variant of the Lagrangian action (1.4), which can be obtained by substituting $\dot{\boldsymbol{x}}=(u,v,0)=(\unicode[STIX]{x1D719}_{x}+\dot{\unicode[STIX]{x1D703}}(y+d_{2}),\unicode[STIX]{x1D719}_{y}-\dot{\unicode[STIX]{x1D703}}(x+d_{1}),0)$ , $\boldsymbol{q}=(q_{1},q_{2},0)$ , $\unicode[STIX]{x1D714}=(0,0,\dot{\unicode[STIX]{x1D703}})$ , $\boldsymbol{d}=(d_{1},d_{2},0)$ , $\overline{\boldsymbol{x}}_{v}=(\overline{x}_{v},\overline{y}_{v},0)$ and

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D64C}=\left[\begin{array}{@{}ccc@{}}\cos \unicode[STIX]{x1D703} & -\text{sin}\unicode[STIX]{x1D703} & 0\\ \sin \unicode[STIX]{x1D703} & \cos \unicode[STIX]{x1D703} & 0\\ 0 & 0 & 1\end{array}\,\right]\end{eqnarray}$$

into (1.4). Then, the variational principle (3.1) reduces to (1.7).

In order to take the variations in (1.7), the variational Reynold’s transport theorem should be used, since the domain of integration $\unicode[STIX]{x1D6FA}$ is time-dependent. See Flanders (Reference Flanders1973), Daniliuk (Reference Daniliuk1976) and Gagarina, van der Vegt & Bokhove (Reference Gagarina, van der Vegt and Bokhove2013) for the background mathematics on the variational analogue of Reynold’s transport theorem. Then, according to the usual procedure in the calculus of variations, equation (1.7) for all variations in the free-surface elevation $\unicode[STIX]{x1D6E4}$ , the velocity potential $\unicode[STIX]{x1D6F7}$ , the vessel position $\boldsymbol{q}=(q_{1},q_{2})$ and the vessel orientation $\unicode[STIX]{x1D703}$ becomes

(3.3) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D6FF}{\mathcal{L}}(\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D703},q_{1},q_{2})=\int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}-\big(\unicode[STIX]{x1D6F7}_{t}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}+gY\big)\big|^{Y=\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}\text{d}X\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle +\,\int _{t_{1}}^{t_{2}}\int _{S}P(X,Y,t)(\unicode[STIX]{x1D6FF}\boldsymbol{X}_{s}\boldsymbol{\cdot }\boldsymbol{n})\,\text{d}s\,\text{d}t-\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}(t)}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{t}+\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7})\unicode[STIX]{x1D70C}\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle +\,\int _{t_{1}}^{t_{2}}\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}(x,t)}\left[\unicode[STIX]{x1D719}_{x}(\unicode[STIX]{x1D6FF}\dot{q}_{1}\cos \unicode[STIX]{x1D703}-\dot{q}_{1}\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FF}\dot{q}_{2}\sin \unicode[STIX]{x1D703}+\dot{q}_{2}\cos \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})\right.\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad \left.+\,\unicode[STIX]{x1D719}_{y}(-\unicode[STIX]{x1D6FF}\dot{q}_{1}\sin \unicode[STIX]{x1D703}-\dot{q}_{1}\cos \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FF}\dot{q}_{2}\cos \unicode[STIX]{x1D703}-\dot{q}_{2}\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})\right.\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad \left.+\,(\dot{q}_{1}\unicode[STIX]{x1D6FF}\dot{q}_{1}+\dot{q}_{2}\unicode[STIX]{x1D6FF}\dot{q}_{2})-g(\cos \unicode[STIX]{x1D703}(x+d_{1})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}-\sin \unicode[STIX]{x1D703}(y+d_{2})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FF}q_{2})\right]\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle +\,\int _{t_{1}}^{t_{2}}\left(m_{v}(\dot{q}_{1}\unicode[STIX]{x1D6FF}\dot{q}_{1}+\dot{q}_{2}\unicode[STIX]{x1D6FF}\dot{q}_{2})-m_{v}\overline{y}_{v}\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D703}}(\dot{q}_{1}\cos \unicode[STIX]{x1D703}+\dot{q}_{2}\sin \unicode[STIX]{x1D703})\right.\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad \left.-\,m_{v}\overline{y}_{v}\dot{\unicode[STIX]{x1D703}}\left(\unicode[STIX]{x1D6FF}\dot{q}_{1}\cos \unicode[STIX]{x1D703}-\dot{q}_{1}\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FF}\dot{q}_{2}\sin \unicode[STIX]{x1D703}+\dot{q}_{2}\cos \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle \left.+\,m_{v}\overline{x}_{v}\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D703}}(-\dot{q}_{1}\sin \unicode[STIX]{x1D703}+\dot{q}_{2}\cos \unicode[STIX]{x1D703})+m_{v}\overline{x}_{v}\dot{\unicode[STIX]{x1D703}}\right.\nonumber\\ \displaystyle & & \displaystyle \quad \times \,(-\unicode[STIX]{x1D6FF}\dot{q}_{1}\sin \unicode[STIX]{x1D703}-\dot{q}_{1}\cos \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FF}\dot{q}_{2}\cos \unicode[STIX]{x1D703}-\dot{q}_{2}\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad \left.+\,m_{v}(\overline{x}_{v}^{2}+\overline{y}_{v}^{2})\dot{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D703}}-m_{v}g(\overline{x}_{v}\cos \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}-\overline{y}_{v}\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FF}q_{2})\right)\text{d}t=0,\end{eqnarray}$$

where

(3.4) $$\begin{eqnarray}P(X,Y,t)=-\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6F7}_{t}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}+gY)\quad \text{on }S\end{eqnarray}$$

and it should be noted that these variations are subject to the restrictions that they vanish at the end points of the time interval and on the vertical boundary at infinity $\unicode[STIX]{x1D6F4}$ . In (3.3), $\boldsymbol{X}_{s}$ denotes the position of a point on the wetted vessel surface $S$ relative to the spatial frame $(X,Y)$ and $\boldsymbol{n}$ is the unit normal vector along $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\supset S$ . The change in $\boldsymbol{X}_{s}$ due to variations in $\boldsymbol{q}$ and $\unicode[STIX]{x1D703}$ is given by

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\boldsymbol{X}_{s}=\unicode[STIX]{x1D64C}^{\prime }\boldsymbol{x}_{s}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}+\unicode[STIX]{x1D6FF}\boldsymbol{q}\quad \text{with }\unicode[STIX]{x1D64C}^{\prime }=\left[\begin{array}{@{}ccc@{}}-\text{sin}\unicode[STIX]{x1D703} & -\text{cos}\unicode[STIX]{x1D703} & 0\\ \cos \unicode[STIX]{x1D703} & -\text{sin}\unicode[STIX]{x1D703} & 0\\ 0 & 0 & 0\end{array}\,\right],\end{eqnarray}$$

where $\boldsymbol{x}_{s}$ is the position of a point on the wetted vessel surface relative to the body frame $(x,y)$ . Taking into account the motion of $\unicode[STIX]{x1D6FA}(t)$ , we may write

(3.6) $$\begin{eqnarray}\displaystyle \displaystyle -\frac{\text{d}}{\text{d}t}\displaystyle \int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t & = & \displaystyle -\displaystyle \int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}\unicode[STIX]{x1D6E4}_{t}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\big|^{Y=\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D70C}\,\text{d}X\,\text{d}t-\int _{t_{1}}^{t_{2}}\int _{S}(\dot{\boldsymbol{X}}_{s}\boldsymbol{\cdot }\boldsymbol{n})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}s\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle -\,\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{t}\unicode[STIX]{x1D70C}\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t.\end{eqnarray}$$

This is the same as the variational Reynold’s transport theorem but with variational derivatives replaced by time derivatives. Noting that the left-hand side vanishes due to the restriction $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}=0$ at times $t=t_{1}$ and $t=t_{2}$ , this expression simplifies to

(3.7) $$\begin{eqnarray}\displaystyle \displaystyle -\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{t}\unicode[STIX]{x1D70C}\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t=\displaystyle \int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}\unicode[STIX]{x1D6E4}_{t}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\big|^{Y=\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D70C}\,\text{d}X\,\text{d}t+\int _{t_{1}}^{t_{2}}\int _{S}(\dot{\boldsymbol{X}}_{s}\boldsymbol{\cdot }\boldsymbol{n})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}s\,\text{d}t. & & \displaystyle\end{eqnarray}$$

With Green’s first identity, we may write

(3.8) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t=\displaystyle -\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t+\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\boldsymbol{\cdot }\boldsymbol{n})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}s\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad =-\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t+\int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}(-\unicode[STIX]{x1D6E4}_{X}\unicode[STIX]{x1D6F7}_{X}+\unicode[STIX]{x1D6F7}_{Y})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\big|^{Y=\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D70C}\,\text{d}X\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle \qquad +\,\int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}(\unicode[STIX]{x1D6F7}_{X}H_{X}+\unicode[STIX]{x1D6F7}_{Y})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\big|_{Y=-H}\unicode[STIX]{x1D70C}\,\text{d}X\,\text{d}t+\int _{t_{1}}^{t_{2}}\int _{S}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}n}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}s\,\text{d}t.\end{eqnarray}$$

Now, using the expressions (3.5), (3.7) and (3.8), integrating by parts and noting that $\unicode[STIX]{x1D6FF}\boldsymbol{q}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ vanish at the end points of the time interval, the variational principle (3.3) simplifies to

(3.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D6FF}{\mathcal{L}}(\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D703},q_{1},q_{2})=\left.\int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}-\left(\unicode[STIX]{x1D6F7}_{t}+\frac{1}{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}+gY\right)\right|^{Y=\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}\text{d}X\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}(\unicode[STIX]{x1D6E4}_{t}+\unicode[STIX]{x1D6E4}_{X}\unicode[STIX]{x1D6F7}_{X}-\unicode[STIX]{x1D6F7}_{Y})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\big|^{Y=\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D70C}\,\text{d}X\,\text{d}t+\int _{t_{1}}^{t_{2}}\int _{S}\left(\dot{\boldsymbol{X}}_{s}\boldsymbol{\cdot }\boldsymbol{n}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}n}\right)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}s\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\int _{t_{1}}^{t_{2}}\int _{X_{1}}^{X_{2}}(\unicode[STIX]{x1D6F7}_{X}H_{X}+\unicode[STIX]{x1D6F7}_{Y})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\big|_{Y=-H}\unicode[STIX]{x1D70C}\,\text{d}X\,\text{d}t+\int _{t_{1}}^{t_{2}}\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}\unicode[STIX]{x1D70C}\,\text{d}\unicode[STIX]{x1D6FA}\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,\int _{t_{1}}^{t_{2}}\int _{S}P(X,Y,t)(\unicode[STIX]{x1D64C}^{\prime }\boldsymbol{x}_{s}\boldsymbol{\cdot }\boldsymbol{n})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\unicode[STIX]{x1D70C}\,\text{d}s\,\text{d}t+\int _{t_{1}}^{t_{2}}\int _{S}P(X,Y,t)\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{q}\unicode[STIX]{x1D70C}\,\text{d}s\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,\int _{t_{1}}^{t_{2}}\bigg(\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}} (-\unicode[STIX]{x1D719}_{xt}\cos \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{x}\dot{\unicode[STIX]{x1D703}}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{yt}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{y}\dot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D703}-\ddot{q}_{1}\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,(\unicode[STIX]{x1D719}_{x}+\dot{\unicode[STIX]{x1D703}}(y+d_{2}))(\unicode[STIX]{x1D719}_{yx}\sin \unicode[STIX]{x1D703}-\unicode[STIX]{x1D719}_{xx}\cos \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,(\unicode[STIX]{x1D719}_{y}-\dot{\unicode[STIX]{x1D703}}(x+d_{1}))(\unicode[STIX]{x1D719}_{yy}\sin \unicode[STIX]{x1D703}-\unicode[STIX]{x1D719}_{xy}\cos \unicode[STIX]{x1D703})\!)\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,m_{v}\ddot{q}_{1}+m_{v}\overline{y}_{v}(\ddot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D703}-\dot{\unicode[STIX]{x1D703}}^{2}\sin \unicode[STIX]{x1D703})+m_{v}\overline{x}_{v}(\ddot{\unicode[STIX]{x1D703}}\sin \unicode[STIX]{x1D703}+\dot{\unicode[STIX]{x1D703}}^{2}\cos \unicode[STIX]{x1D703})\bigg)\unicode[STIX]{x1D6FF}q_{1}\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle +\,\int _{t_{1}}^{t_{2}}\bigg(\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}} (-\unicode[STIX]{x1D719}_{xt}\sin \unicode[STIX]{x1D703}-\unicode[STIX]{x1D719}_{x}\dot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D703}-\unicode[STIX]{x1D719}_{yt}\cos \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{y}\dot{\unicode[STIX]{x1D703}}\sin \unicode[STIX]{x1D703}-\ddot{q}_{2}-g\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,(\unicode[STIX]{x1D719}_{xx}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{yx}\cos \unicode[STIX]{x1D703})(\unicode[STIX]{x1D719}_{x}+\dot{\unicode[STIX]{x1D703}}(y+d_{2}))\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,(\unicode[STIX]{x1D719}_{xy}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{yy}\cos \unicode[STIX]{x1D703})(\unicode[STIX]{x1D719}_{y}-\dot{\unicode[STIX]{x1D703}}(x+d_{1}))\!)\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle -\,m_{v}\ddot{q}_{2}+m_{v}\overline{y}_{v}(\ddot{\unicode[STIX]{x1D703}}\sin \unicode[STIX]{x1D703}+\dot{\unicode[STIX]{x1D703}}^{2}\cos \unicode[STIX]{x1D703})-m_{v}\overline{x}_{v}(\ddot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D703}-\dot{\unicode[STIX]{x1D703}}^{2}\sin \unicode[STIX]{x1D703})-m_{v}g\bigg)\unicode[STIX]{x1D6FF}q_{2}\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle \quad \displaystyle +\,\int _{t_{1}}^{t_{2}}\bigg(\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}}\left[\unicode[STIX]{x1D719}_{x}(-\dot{q}_{1}\sin \unicode[STIX]{x1D703}+\dot{q}_{2}\cos \unicode[STIX]{x1D703})+\unicode[STIX]{x1D719}_{y}(-\dot{q}_{1}\cos \unicode[STIX]{x1D703}-\dot{q}_{2}\sin \unicode[STIX]{x1D703})\right.\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad \left.\left.-\,g(\cos \unicode[STIX]{x1D703}(x+d_{1})-\sin \unicode[STIX]{x1D703}(y+d_{2}))\right]\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x+m_{v}\overline{y}_{v}(\ddot{q}_{1}\cos \unicode[STIX]{x1D703}+\ddot{q}_{2}\sin \unicode[STIX]{x1D703})\right.\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,m_{v}\overline{x}_{v}(-\ddot{q}_{1}\sin \unicode[STIX]{x1D703}+\ddot{q}_{2}\cos \unicode[STIX]{x1D703})-m_{v}(\overline{x}_{v}^{2}+\overline{y}_{v}^{2})\ddot{\unicode[STIX]{x1D703}}-m_{v}g(\overline{x}_{v}\cos \unicode[STIX]{x1D703}-\overline{y}_{v}\sin \unicode[STIX]{x1D703})\bigg)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\,\text{d}t=0.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

From (3.9), we conclude that invariance of ${\mathcal{L}}$ with respect to a variation in the free-surface elevation $\unicode[STIX]{x1D6E4}$ yields the dynamic free-surface boundary condition in (1.1), invariance of ${\mathcal{L}}$ with respect to a variation in the velocity potential $\unicode[STIX]{x1D6F7}$ yields the field equation in $\unicode[STIX]{x1D6FA}(t)$ , invariance of ${\mathcal{L}}$ with respect to a variation in the velocity potential $\unicode[STIX]{x1D6F7}$ at $Y=-H(X)$ gives the bottom boundary condition in (1.1), invariance of ${\mathcal{L}}$ with respect to a variation in the velocity potential $\unicode[STIX]{x1D6F7}$ at $Y=\unicode[STIX]{x1D6E4}(X,t)$ gives the kinematic free-surface boundary condition in (1.1) and invariance of ${\mathcal{L}}$ with respect to a variation in the velocity potential $\unicode[STIX]{x1D6F7}$ on $S$ gives the contact condition on the wetted surface of the vessel,

(3.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}}{\unicode[STIX]{x2202}n}=\dot{\boldsymbol{X}}_{s}\boldsymbol{\cdot }\boldsymbol{n}\quad \text{on }S.\end{eqnarray}$$

Finally, invariance of ${\mathcal{L}}$ with respect to variations in $q_{1}$ , $q_{2}$ and $\unicode[STIX]{x1D703}$ gives the hydrodynamic equations of motion for the rigid body in the sway/surge $(q_{1})$ , heave $(q_{2})$ and roll/pitch $(\unicode[STIX]{x1D703})$ directions respectively, which are

1. (i)

   (3.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle (m_{v}+m_{f})\ddot{q}_{1}-\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}} (-\unicode[STIX]{x1D719}_{xt}\cos \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{yt}\sin \unicode[STIX]{x1D703}+\dot{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D719}_{x}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{y}\cos \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,(\unicode[STIX]{x1D719}_{x}+\dot{\unicode[STIX]{x1D703}}(y+d_{2}))(\unicode[STIX]{x1D719}_{yx}\sin \unicode[STIX]{x1D703}-\unicode[STIX]{x1D719}_{xx}\cos \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,(\unicode[STIX]{x1D719}_{y}-\dot{\unicode[STIX]{x1D703}}(x+d_{1}))(\unicode[STIX]{x1D719}_{yy}\sin \unicode[STIX]{x1D703}-\unicode[STIX]{x1D719}_{xy}\cos \unicode[STIX]{x1D703})\!)\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,m_{v}\overline{y}_{v}(\ddot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D703}-\dot{\unicode[STIX]{x1D703}}^{2}\sin \unicode[STIX]{x1D703})-m_{v}\overline{x}_{v}(\ddot{\unicode[STIX]{x1D703}}\sin \unicode[STIX]{x1D703}+\dot{\unicode[STIX]{x1D703}}^{2}\cos \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,\int _{S}P(X,Y,t)n_{1}\,\text{d}s=0,\qquad\end{eqnarray}$$

2. (ii)

   (3.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle (m_{v}+m_{f})\left(g+\ddot{q}_{2}\right)+\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}} (\unicode[STIX]{x1D719}_{xt}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{yt}\cos \unicode[STIX]{x1D703}+\dot{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D719}_{x}\cos \unicode[STIX]{x1D703}-\unicode[STIX]{x1D719}_{y}\sin \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,(\unicode[STIX]{x1D719}_{xx}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{yx}\cos \unicode[STIX]{x1D703})(\unicode[STIX]{x1D719}_{x}+\dot{\unicode[STIX]{x1D703}}(y+d_{2}))\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,(\unicode[STIX]{x1D719}_{xy}\sin \unicode[STIX]{x1D703}+\unicode[STIX]{x1D719}_{yy}\cos \unicode[STIX]{x1D703})(\unicode[STIX]{x1D719}_{y}-\dot{\unicode[STIX]{x1D703}}(x+d_{1}))\!)\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,m_{v}\overline{y}_{v}(\ddot{\unicode[STIX]{x1D703}}\sin \unicode[STIX]{x1D703}+\dot{\unicode[STIX]{x1D703}}^{2}\cos \unicode[STIX]{x1D703})+m_{v}\overline{x}_{v}(\ddot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D703}-\dot{\unicode[STIX]{x1D703}}^{2}\sin \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,\int _{S}P(X,Y,t)n_{2}\,\text{d}s=0,\qquad\end{eqnarray}$$

3. (iii)

   (3.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle m_{v}(\overline{x}_{v}^{2}+\overline{y}_{v}^{2})\ddot{\unicode[STIX]{x1D703}}-\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}} [\unicode[STIX]{x1D719}_{x}(-\dot{q}_{1}\sin \unicode[STIX]{x1D703}+\dot{q}_{2}\cos \unicode[STIX]{x1D703})+\unicode[STIX]{x1D719}_{y}(-\dot{q}_{1}\cos \unicode[STIX]{x1D703}-\dot{q}_{2}\sin \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,g(\cos \unicode[STIX]{x1D703}(x+d_{1})-\sin \unicode[STIX]{x1D703}(y+d_{2}))]\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x-m_{v}\overline{y}_{v}(\ddot{q}_{1}\cos \unicode[STIX]{x1D703}+\ddot{q}_{2}\sin \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad +\,m_{v}\overline{x}_{v}(-\ddot{q}_{1}\sin \unicode[STIX]{x1D703}+\ddot{q}_{2}\cos \unicode[STIX]{x1D703})+m_{v}g(\overline{x}_{v}\cos \unicode[STIX]{x1D703}-\overline{y}_{v}\sin \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad -\,\int _{S}P(X,Y,t)(\unicode[STIX]{x1D64C}^{\prime }\boldsymbol{x}_{s}\boldsymbol{\cdot }\boldsymbol{n})\,\text{d}s=0,\end{eqnarray}$$

where $P(X,Y,t)$ is defined in (3.4) and $m_{f}=\int _{0}^{L}\int _{0}^{\unicode[STIX]{x1D702}}\unicode[STIX]{x1D70C}\,\text{d}y\,\text{d}x$ is independent of time.

In summary, the equations of motion for the exterior water waves in $\unicode[STIX]{x1D6FA}(t)$ are (1.1) with the contact boundary condition (3.10). The equations of motion for the interior fluid motion are the field equation (2.5) with the boundary conditions (2.7)–(2.9) which are dynamically coupled to the hydrodynamic equations of motion for the rigid body (3.11)–(3.13). The terms including the pressure field $P(X,Y,t)$ in the $q_{1},q_{2}$ equations and also in the $\unicode[STIX]{x1D703}$ equation in (3.11)–(3.13) are the forces and moment respectively acting on the rigid body due to the exterior water waves. Similarly, the integral terms including the derivatives of $\unicode[STIX]{x1D719}(x,y,t)$ are the forces and moment acting on the rigid body due to the interior fluid motion.

4 Concluding remarks

This paper is devoted to the derivation of new variational principles for 2D interactions between water waves and a rigid body with interior fluid sloshing. The complete set of equations of motion and boundary conditions for the exterior water waves and for the interior fluid motion of the vessel relative to the rotating–translating coordinate system attached to the moving vessel, and also the exact nonlinear Euler–Lagrange equations for the rigid-body motion in the sway/surge, heave and roll/pitch directions, are derived from the presented variational principles. The proposed variational principles are applicable to ocean engineering problems. The exact differential equations can be used for the coupled dynamical analysis of a freely floating ship with water on deck or with interior fluid sloshing in the tanks which interacts with exterior water waves.

Another interesting application of the presented coupled variational principles is for the dynamical response analysis of floating ocean wave energy converters (WECs) such as the OWEL wave energy converter proposed by Offshore Wave Energy Ltd, a schematic of which can be found on the website http://www.owel.co.uk/. OWEL is a floating vessel with variable topography and cross-section, open at one end to capture ocean waves. Once they are trapped, the waves undergo interior fluid sloshing while the vessel is interacting with exterior waves. A rise in the wave height is induced within the duct, mainly due to the internal geometry of the WEC. The wave then creates a seal with the rigid lid, resulting in a moving trapped pocket of air ahead of the wave front which drives the power take-off.

The proposed variational principles can be used for mathematical modelling of the pendulum–slosh problem. The rigid-body equation for the pendular motion of a rectangular vessel suspended from a single rigid pivoting rod, partially filled with an inviscid fluid, can be derived from a simplified version of the variational principle (1.7). One can consider the second part of (1.7) with $q_{1}=q_{2}=0$ and take the variation with respect to $\unicode[STIX]{x1D703}$ to obtain the rigid-body equation. The complete set of equations of motion for the interior fluid of the pendulum can be obtained by setting $q_{1}=q_{2}=0$ in (1.8) and taking the variations with respect to $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D702}$ .

A direction of great interest is to use the variational symplectic methods of Gagarina et al. (Reference Gagarina, Ambati, van der Vegt and Bokhove2014, Reference Gagarina, Ambati, Nurijanyan, van der Vegt and Bokhove2016), Bokhove & Kalogirou (Reference Bokhove, Kalogirou, Bridges, Groves and Nicholls2016) and Kalogirou & Bokhove (Reference Kalogirou and Bokhove2016) to develop energy preserving numerical solvers for the proposed variational principles (1.7) and (1.8) for interactions between ocean waves and floating structures dynamically coupled to interior fluid sloshing, and also for the variational principle (1.8) for the problem of fluid sloshing in vessels undergoing prescribed rigid-body motion in two dimensions.