Classical and quantum dynamics for an extended free rigid body

In this paper, an extended free rigid body of dimension three is defin
ed and analysed both in classical and quantum mechanics. The inverse
inertia tensor, which is a positive-definite symmetric matrix in the
ordinary case, is generalized to arbitrary symmetric ones in our
case. With an arbitrary symmetric matrix chosen, associated is a
Lie-Poisson structure on the Euclidean space of dimension three,
through which the classical dynamics for an extended free rigid body
is analysed by means of two characteristic first integrals. Further,
the flows on the energy surfaces, their fixed points, and their
stabili ty are studied. In parrallel with this, the quantum dynamics
is formulated as the problem of the simultaneous spectral resolution
of the two operators which are viewed as the q uantization of the
first integrals in the classical dynamics of the extended fre e rigid
body. The explicit spectral resolution are obtained, when the extended
rigid body is a n extended symmetric top.