Scale relativistic formulation of non-differentiable mechanics I: Application to the harmonic oscillator

This article is the first in a series of two presenting the scale relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. In this first paper, we present the definition of 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 reveals the same statistics as the stationary solutions of the Schroedinger equation for the same problem. This motivates the second paper which makes the relation to quantum mechanics explicit by discussing the axioms of quantum mechanics in the Scale Relativity framework.