Integrable problems of the dynamics of coupled rigid bodies

Several classical problems of dynamics are shown to be integrable for the special systems of coupled rigid bodies introduced in this paper and called $C^k$-central configurations. It is proved that the dynamics of an arbitrary $C^k$-central configuration in the Newtonian gravitational field with an arbitrary quadratic potential is integrable in the Liouville sense and in theta-functions of Riemann surfaces. A hidden symmetry of the inertial dynamics of these configurations is found, and reductions of the corresponding Lagrange equations to the Euler equations on the direct sums of Lie coalgebras $SO(3)$ are obtained. Reductions and integrable cases of the equations for the rotation of a heavy $C^k$-central configuration about a fixed point are indicated. Separation of rotations of a space station type orbiting system, which is a $C^k$-central configuration of rigid bodies, is proved. This result leads to the possibility of independent stabilization of rotations of the rigid bodies in such orbiting configurations. Integrability of the inertial dynamics of $CR^n$-central configurations of coupled gyrostats is proved.