Kirchhoff's equations

In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid.

where ${\displaystyle {\vec {\omega }}}$ and ${\displaystyle {\vec {v}}}$ are the angular and linear velocity vectors at the point ${\displaystyle {\vec {x}}}$, respectively; ${\displaystyle {\tilde {I}}}$ is the moment of inertia tensor, ${\displaystyle m}$ is the body's mass; ${\displaystyle {\hat {n}}}$ is a unit normal to the surface of the body at the point ${\displaystyle {\vec {x}}}$; ${\displaystyle p}$ is a pressure at this point; ${\displaystyle {\vec {Q}}_{h}}$ and ${\displaystyle {\vec {F}}_{h}}$ are the hydrodynamic torque and force acting on the body, respectively; ${\displaystyle {\vec {Q}}}$ and ${\displaystyle {\vec {F}}}$ likewise denote all other torques and forces acting on the body. The integration is performed over the fluid-exposed portion of the body's surface.

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