  This article motivates and presents the scale relativistic approach to
non-differentiability in mechanics and its relation to quantum mechanics. It
stems from the scale relativity proposal to extend the principle of relativity
to resolution-scale transformations, which leads to considering
non-differentiable dynamical paths. We first define a complex scale-covariant
time-differential operator and show that mechanics of non-differentiable paths
is implemented in the same way as classical mechanics but with the replacement
of the time derivative and velocity with the time-differential operator and
associated complex velocity. With this, the generalized form of Newton's
fundamental relation of dynamics is shown to take the form of a Langevin
equation in the case of stationary motion characterized by a null average
classical velocity. The numerical integration of the Langevin equation in the
case of a harmonic oscillator taken as an example reveals the same statistics
as the stationary solutions of the Schrodinger equation for the same problem.
This motivates the rest of the paper, which shows Schrodinger's equation to be
a reformulation of Newton's fundamental relation of dynamics as generalized to
non-differentiable geometries and leads to an alternative interpretation of the
other axioms of standard quantum mechanics in a coherent picture. This exercise
validates the scale relativistic approach and, at the same time, it allows to
envision macroscopic chaotic systems observed at resolution time-scales
exceeding their horizon of predictability as candidates in which to search for
quantum-like dynamics and structures.
