I found this paper[1] about the fractional variational principle.

I didn’t know anything about fractional derivative/integration before taking a look at this paper. If you like to know more, you can check here. It gives the basic ideas.

Beside this fractional concept, there is a nice thing in the introduction of the paper which may be apparent to any physicist, but I didn’t know it:

“For every variational symmetry of the problem, there corresponds a conservation law”.

For instance, spatial invariance is equivalent to the conservation of linear momentum, and the time invariance is equivalent to the conservation of energy.

I’m wondering what does the invariance of a function w.r.t. choice of coordinate system mean (in the sense of having the same function defined on different charting of a manifold)? Some relativistic property?

[1] G. S. F. Frederico and D. F. M. Torres, “Fractional Optimal Control in the Sense of Caputo and the Fractional Noether’s Theorem,” Dec 2007, preprint.

Invariance w.r.t. local coordinates means nothing: according to one principle of Relativity, Physics should not depend on any peculiar choice of coordinates. Therefore, from the very beginning, physicists (and differential geometers) restrict their interest to “tensors” (objects that are invariant, by construction, to changes of coordinates). Therefore, the question that you ask does not arise.

Thanks! You enlightened me.