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Integrable problems of the dynamics of coupled rigid bodies
O. I. Bogoyavlenskii
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.
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Russian Academy of Sciences. Izvestiya Mathematics, 1993, 41:3, 395–416
MSC: Primary 70F99, 70H35; Secondary 70M20
O. I. Bogoyavlenskii, “Integrable problems of the dynamics of coupled rigid bodies”, Izv. RAN. Ser. Mat., 56:6 (1992), 1139–1164; Russian Acad. Sci. Izv. Math., 41:3 (1993), 395–416
Citation in format AMSBIB
\paper Integrable problems of the dynamics of coupled rigid bodies
\jour Izv. RAN. Ser. Mat.
\jour Russian Acad. Sci. Izv. Math.
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Shamolin M.V., “Variety of the Cases of Integrability in Dynamics of a Symmetric 2D-, 3D-and 4D-Rigid Body in a Nonconservative Field”, Int. J. Struct. Stab. Dyn., 13:7, SI (2013), 1340011
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