  There are at least three different notions of degrees of freedom (DF) that
are important in comparison of quantum and classical dynamical systems. One is
related to the type of dynamical equations and inequivalent initial conditions,
the other to the structure of the system and the third to the properties of
dynamical orbits. In this paper, definitions and comparison in classical and
quantum systems of the tree types of DF are formulated and discussed. In
particular, we concentrate on comparison of the number of the so called
dynamical DF in a quantum system and its classical model. The comparison
involves analyzes of relations between integrability of the classical model,
dynamical symmetry and separability of the quantum and the corresponding
classical systems and dynamical generation of appropriately defined
quantumness. The analyzes is conducted using illustrative typical systems. A
conjecture summarizing the observed relation between generation of quantumness
by the quantum dynamics and dynamical properties of the classical model is
formulated.
