On the KAM - Theory Conditions for the Kirchhoff Top

Christov, Ognyan

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The present paper deals with the KAM-theory conditions for systems describing the motion of a particle in central field.

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* Partialy supported by contract MM 523/95 with Ministry of Science and Technologies of Republic of Bulgaria. For the system describing the motion of a moss point under the action of two static gravity centers (with equal masses), we find a subset of the set of the regular values of the energy and momentum, where the condition of isoenergetical non-degeneracy is fulfilled.

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We discuss the motion of the three-dimensional rigid body about a fixed point under the influence of gravity, more specifically from the point of view of its symplectic structures and its constants of the motion. An obvious symmetry reduces the problem to a Hamiltonian flow on a four-dimensional submanifold of $s o (3) \times s o (3)$ ; they are the customary Euler-Poisson equations. This symplectic manifold can also be regarded as a coadjoint orbit of the Lie algebra of the semi-direct product group $S O (3) \times s o (3)$ with...