Kirchhoff equations

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Motion of a rigid body in an ideal fluid in projections onto the axes of the body fixed frame $C x 1 x 2 x 3$ can be described by the equations

${d\over{dt}} {{\partial T}\over{\partial \omega}} = {{\partial T}\over{\partial \omega}} \times \omega + {{\partial T}\over{\partial v}} \times v + Q_h + Q, \quad {d\over{dt}} {{\partial T}\over{\partial v}} = {{\partial T}\over{\partial v}} \times \omega + F_h + F, \quad T = {1 \over 2} \left( \omega^T I \omega + m v^2 \right)$

$Q_h=-\int p(x) x \times n(x) d\sigma(x), \quad F_h=-\int p(x) n(x) d\sigma(x)$

where $ω$ and $v$ are vectors of body's angular velocity and the velocity of the point $C$ respectively; $I$ and $m$ are a central tensor of inertia of the body and its mass; $n (x)$ is an external normal to the surface of the body at the point $x$ ; $p (x)$ is a pressure at this point; $Q h$ and $F h$ are the torque and the force acting at the body from the side of fluid; $Q$ and $F$ denote all other torques and forces acting at the body. The integration is realized over the moistened part of body's surface.

If the completely submerged body moves in an infinitely large volume of irrotational, incompressible, inviscid fluid, that is at rest at infinity, then the vectors $Q h$ and $F h$ can be found via explicit integration, and the dynamics of the body is described by Kirchhoff - Clebsch equations

${d\over{dt}} {{\partial L}\over{\partial \omega}} = {{\partial L}\over{\partial \omega}} \times \omega + {{\partial L}\over{\partial v}} \times v, \quad {d\over{dt}} {{\partial L}\over{\partial v}} = {{\partial L}\over{\partial v}} \times \omega,$

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

Their first integrals read

$J_0 = \left({{\partial L}\over{\partial \omega}}, \omega \right) + \left({{\partial L}\over{\partial v}}, v \right) - L, \quad J_1 = \left({{\partial L}\over{\partial \omega}} {{\partial L}\over{\partial v}}\right), \quad J_2 = \left({{\partial L}\over{\partial v}} {{\partial L}\over{\partial v}}\right)$