Kirchhoff equations

The motion of a rigid body in an ideal fluid can be expressed in a basis fixed to the body by Kirchhoff's equations:

$\{dover\{dt$partial T}over{partial vec omega = partialT}over{partial vec omega imes vec omega + partialT}over{partial vec v imes vec v + vec Q_h + vec Q, quad{dover{dt partial T}over{partial vec v = partialT}over{partial vec v imes vec omega + vec F_h + vec F, quad T = {1 over 2} left( vec omega^T ilde I vec omega + m v^2 ight)

$vec\; Q\_h=-int\; p\; vec\; x\; imes\; hat\; n\; dsigma,\; quadvec\; F\_h=-int\; p\; hat\; n\; dsigma$

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

If the body is completely submerged body in an infinitely largevolume of irrotational, incompressible, inviscid fluid, that is atrest at infinity, then the vectors $vec\; Q\_h$ and $vec\; F\_h$ can befound via explicit integration, and the dynamics of the body isdescribed by the Kirchhoff - Clebsch equations:

$\{dover\{dt$partial L}over{partial vec v = partial L}over{partial vec v imes vec omega,

$L(vec\; omega,\; vec\; v)\; =\; \{1\; over\; 2\}\; (A\; vec\; omega,vec\; omega)\; +\; (B\; vec\; omega,vec\; v)\; +\; \{1\; over\; 2\}\; (C\; vec\; v,vec\; v)\; +\; (vec\; k,vec\; omega)\; +\; (vec\; l,vec\; v).$

Their first integrals read

$J\_0\; =\; left($partial L}over{partial vec omega, vec omega ight) + left(partial L}over{partial vec v, vec v ight) - L, quadJ_1 = left(partial L}over{partial vec omega,partial L}over{partial vec v ight), quad J_2 = left(partial L}over{partial vec v,partial L}over{partial vec v ight) .

Further integration produces explicit expressions for position and velocities.

References

* Kirchhoff G. R. Vorlesungen ueber Mathematische Physik, Mechanik. Lecture 19. Leipzig: Teubner. 1877.
* Lamb, H. - Hydrodynamics. Sixth Edition Cambridge (UK): Cambridge University Press. 1932.