Kirchhoff equations

In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid.
where formula_2 and formula_3 are the angular and linear velocity vectors at the point formula_4, respectively; formula_5 is the moment of inertia tensor, formula_6 is the body's mass; formula_7 is
a unit normal to the surface of the body at the point formula_4;
formula_9 is a pressure at this point; formula_10 and formula_11 are the hydrodynamic
torque and force acting on the body, respectively;
formula_12 and formula_13 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.
If the body is completely submerged body in an infinitely large volume of irrotational, incompressible, inviscid fluid, that is at rest at infinity, then the vectors formula_10 and formula_11 can be found via explicit integration, and the dynamics of the body is described by the Kirchhoff – Clebsch equations:
Their first integrals read
Further integration produces explicit expressions for position and velocities. 