  Newtonian and Schrodinger dynamics can be formulated in a physically
meaningful way within the same Hilbert space framework. This fact was recently
used to discover an unexpected relation between classical and quantum motions
that goes beyond the results provided by the Ehrenfest theorem. The Newtonian
dynamics was shown to be the Schrodinger dynamics of states constrained to a
submanifold of the space of states, identified with the classical phase space
of the system. Quantum observables are identified with vector fields on the
space of states. The commutators of observables are expressed through the
curvature of the space. The resulting embedding of the Newtonian and
Schrodinger dynamics into a unified geometric framework is rigid in the sense
that the Schrodinger dynamics is a unique extension of the Newtonian one. Under
the embedding, the normal distribution of measurement results associated with a
classical measurement implies the Born rule for the probability of transition
of quantum states. The mathematics of the discovered relationship between the
classical and the quantum is reviewed and investigated here in detail, and
applied to the process of measurement of spin and position observables.
