Non-Differentiable Mechanical Model and Its Implications

Considering that the motions of the particles take place on fractals, a non-differentiable mechanical model is built. Only if the spatial coordinates are fractal functions, the Galilean version of our model is obtained: the geodesics satisfy a Navier-Stokes-type of equation with an imaginary viscosity coefficient for a complex speed field or respectively, a Schrödinger-type of equation or hydrodynamic equations, in the case of irrotational movements. Moreover, in this approach, the analysis of the fractal fluid dynamics generates conductive properties in the case of movements synchronization both on differentiable and fractal scales, and convective properties in the absence of synchronization ( e.g. laser ablation plasma is analyzed). On the other hand, if both the spatial and temporal coordinates are fractal functions, it results that, the geodesics satisfy a Klein-Gordon-type of equation on a Minkowskian manifold.