Kirchhoff equations
In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid. d d t ∂ T ∂ ω = ∂ T ∂ ω × ω + ∂ T ∂ v × v + Q h + Q , d d t ∂ T ∂ v = ∂ T ∂ v × ω + F h + F , T = 1 2 ( ω T I ~ ω + m v 2 ) Q h = − ∫ p x × n ^ d σ , F h = − ∫ p n ^ d σ {\displaystyle {\begin{aligned}{\mathrm {d} \over \mathrm {d} t}{{\partial T} \over {\partial {\boldsymbol {\omega }}}}&={{\partial T} \over {\partial {\boldsymbol {\omega }}}}\times {\boldsymbol {\omega }}+{{\partial T} \over {\partial \mathbf {v} }}\times \mathbf {v} +\mathbf {Q} _{h}+\mathbf {Q} ,\\[10pt]{\mathrm {d} \over \mathrm {d} t}{{\partial T} \over {\partial \mathbf {v} }}&={{\partial T} \over {\partial \mathbf {v} }}\times {\boldsymbol {\omega }}+\mathbf {F} _{h}+\mathbf {F} ,\\[10pt]T&={1 \over 2}\left({\boldsymbol {\omega }}^{T}{\tilde {I}}{\boldsymbol {\omega }}+mv^{2}\right)\\[10pt]\mathbf {Q} _{h}&=-\int p\mathbf {x} \times {\hat {\mathbf {n} }}\,d\sigma ,\\[10pt]\mathbf {F} _{h}&=-\int p{\hat {\mathbf {n} }}\,d\sigma \end{aligned}}} where ω {\displaystyle {\boldsymbol {\omega }}} and v {\displaystyle \mathbf {v} } are the angular and linear velocity vectors at the point x {\displaystyle \mathbf {x} } , respectively; I ~ {\displaystyle {\tilde {I}}} is the moment of inertia tensor, m {\displaystyle m} is the body's mass; n ^ {\displaystyle {\hat {\mathbf {n} }}} is a unit normal vector to the surface of the body at the point x {\displaystyle \mathbf {x} } ; p {\displaystyle p} is a pressure at this point; Q h {\displaystyle \mathbf {Q} _{h}} and F h {\displaystyle \mathbf {F} _{h}} are the hydrodynamic torque and force acting on the body, respectively; Q {\displaystyle \mathbf {Q} } and F {\displaystyle \mathbf {F} } likewise denote all other torques and forces acting on the body.