  In this paper, the classical Schr\"odinger equation, which allows the study
of classical dynamics in terms of wave functions, is analyzed theoretically and
numerically. First, departing from classical (Newtonian) mechanics, and
assuming an additional single-valued condition for the Hamilton's principal
function, the classical Schr\"odinger equation is obtained. This additional
assumption implies inherent non-classical features on the description of the
dynamics obtained from the classical Schr\"odinger equation: the trajectories
do not cross in the configuration space. Second, departing from Bohmian
mechanics and invoking the quantum-to-classical transition, the classical
Schr\"odinger equation is obtained in a natural way for the center of mass of a
quantum system with a large number of identical particles. This quantum
development imposes the condition of dealing with a narrow wave packet, which
implicitly avoids the non-classical features mentioned above. We illustrate all
the above points with numerical simulations of the classical and quantum
Schr\"odinger equations for different systems.
