As far as I know, the only article which explains the physical and geometrical interpretation of fractional derivatives and integrals is by Podlubny (see link below). And, a special case of physical interpretation will be available Monday or Tuesday morning on Arxiv.org. I will be uploading the preprint of my next submission there.

And, for some cool animation of it, look at my youtube channel. (attached below)

http://arxiv.org/pdf/math/0110241.pdf

https://www.youtube.com/channel/UCvLjlCpW1g1bm8vzo-WAyuw/videos

@ Luis Barreira,

The link corresponds to Herrmann's  book and he takes it from Podlubny's article. One of the first physical problem in which fractional calculus was used was the tautochrone problem by Abel, which your link illustrates.

thanks to all

The geometrical and physical meaning of ordinary derivative is simple and intuitive: For smooth function f which is differentiable at x, the local behavior of f around point x can be approximated by using the ordinary derivative: f(x + h) ≈ f(x) + hf′(x). The ordinary derivative gives the linear approximation of smooth function. Here we expect the fractional derivative to have the similar geometrical meaning.

We hope for non-differentiable functions, the fractional derivative could give some kind approximation of its local behavior. A SIMPLE DEFINITION DIRECTLY FROM GEOMETRICAL MEANING: We expect that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions f(x + h) ≈ f(x) + h^α f^(α)(x)/Γ(α + 1) in which the function f is not differentiable because df ≈ (dx)^α so the classical derivative df/dx will diverge. Note that the purpose of adding the coefficient Γ(α + 1) is just to make the formal consistency with the Taylor series.

Fractional dynamics is the field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power law non-locality, power law long-term memory or fractal properties by using integrations and differentiation of non integer orders, i.e., by utilizing methods of the fractional calculus.

Thank you for your answer

This is not an answer, but an expression of results in some previous literature:

The fractional derivative, we can say that it is a promising tool for observing the extent of the correlation or the strength of the bridge linking the function and its derivative along the domain, if we take into account that $\alpha$ is between zero and one, in addition to that this relationship is not local.