Application of Kirchhoff's equations of motion to the dynamically coupled system of a rigid body with a completely-liquid-filled cavity.

This article addresses the dynamics of a rigid body containing a cavity completely filled with an inviscid, incompressible, homogeneous liquid, focusing on the coupled motion of the body and fluid without external forces or torques. It highlights that Kirchhoff’s equations of motion, originally formulated for a rigid body immersed in an external fluid, also apply to this internal flow problem under assumptions of irrotational flow and no external influences. The paper develops the mathematical framework, including expressions for linear and angular momenta, and presents a planar example of a circular cavity with an added point vortex to incorporate vorticity effects, showing that the vortex alters angular momentum but not linear momentum. The system’s dynamics are formulated in a Hamiltonian framework with Lie–Poisson and Poisson brackets, enabling analysis of conserved quantities and stability. The study suggests potential extensions to elastodynamics by replacing the rigid body with an elastic one, linking fluid–structure interaction problems across fluid mechanics and elasticity.