This paper derives an averaged Lagrangian functional for dynamic coupling between rigid-body motion and its interior shallow-water sloshing in three-dimensional rotating and translating coordinates; with a time-dependent rotation vector. A new set of variational shallow-water equations (SWEs) and generalized Green–Naghdi equations for the interior fluid sloshing with 3-D rotation vector and translations, and also the equations of motion for the linear momentum and angular momentum of the rigid-body containing shallow water, are derived from the averaged Lagrangian functional, which describes a columnar motion, by using Hamilton’s principle and the Euler–Poincaré variational framework. The generalized Green–Naghdi equations have a form of potential vorticity (PV) conservation, which can be obtained from the particle-relabelling symmetry, and is a combination of the PV derived by Miles and Salmon (1985) and the PV derived by Dellar and Salmon (2005) for geophysical fluid dynamics problems, where the rotation vector varies spatially. By applying the assumption of zero-potential-vorticity flow to the averaged Lagrangian functional, a new set of Boussinesq-like evolution equations are derived, which are a generalization of the Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates. Moreover, the new variational principles are appended to Luke’s variational principle to present a unified variational framework for the hydrodynamic problem of interactions between gravity-driven potential-flow water waves and a freely floating rigid-body, dynamically coupled to its interior weakly dispersive nonlinear shallow-water sloshing in three dimensions.

本文推导了一个平均的拉格朗日泛函，用于在三维旋转和平移坐标中对刚体运动与其内部浅水晃动进行动态耦合。与时间相关的旋转矢量。一组新的变分浅水方程组（SWE）和广义Green-Naghdi方程组，用于带有3-D旋转矢量和平移的内部流体晃动，以及刚体的线性动量和角动量的运动方程包含浅水，是从拉格朗日平均函数得出的，该函数描述了柱状运动，利用汉密尔顿原理和欧拉-庞加莱变分框架。广义的Green–Naghdi方程具有一种形式的潜在涡度（PV）守恒，可以从粒子重新标记对称性获得，它是Miles和Salmon（1985）推导的PV与Dellar和Salmon（2005）研究地球物理流体动力学问题，其中旋转矢量在空间上变化。通过将零势涡流的假设应用于平均拉格朗日泛函，推导了一组新的类Boussinesq演化方程，该方程是在三维旋转和平移坐标系中流体晃动的Whitham方程的推广。而且，