  With an apparent delay of over one century with respect to the development of
standard Analytical Mechanics, but still in fully classical terms, the behavior
of classical monochromatic wave beams in stationary media is shown to be ruled
by a dispersive "Wave Potential" function, encoded in the structure of the
Helmholtz equation. An exact, ray-based Hamiltonian description, revealing a
strong ray coupling due to the Wave Potential, and reducing to the geometrical
optics approximation when this function is neglected, is shown to hold even for
typically wave-like phenomena such as diffraction and interference. Recalling,
then, that the time-independent Schroedinger equation (associating the quantum
motion of mono-energetic particles with stationary monochromatic matter waves)
is itself a Helmholtz-like equation, the mathematical treatment holding in the
classical case is extended, without resorting to statistical concepts, to the
exact, trajectory-based, Hamiltonian quantum dynamics of point-like particles.
The particle trajectories and motion laws turn out to be coupled, in this case,
by a function strictly analogous to the Wave Potential and formally assuming
the familiar form of Bohm's "Quantum Potential", which is therefore not so much
a "quantum" as a "wave" property - in whose absence the quantum particle
dynamics reduces to the classical one. The time-independent Schroedinger
equation is argued to be not a trivial particular case of the time-dependent
one, but the exact quantum dynamical ground on which Schroedinger's
time-dependent statistical description (representing particles as travelling
wave-packets) is based. It provides indeed the (exact) link between classical
particle dynamics and Bohm's hydrodynamics.
