Undergrad Degrees of Freedom in Lagrangian Mechanics for a Fractal Path

[Pikkugnome]

TL;DR

Degree of freedom along a strange curve from lagrangian mechanics point of view.

Degree of freedom along a parabola, or any such tame curve, is one from lagrangian mechanics point of view. It makes sense. However how does degree of freedom accompany a space filling curve. Intuitively degree of freedom is not two, since not all motions are possible along the curve. How would lagrangian mechanics work on a such curve. A fractal can have a non integer dimension, would that make its degree of freedom also a non integer.

 

[pasmith]

How does Lagrangian mechanics work if motion is confined to a curve which is nowhere differentiable, as a fractal curve almost certainly is?

 

[.Scott]

That's an interesting question.

The boundary to the Mandelbrot Set has a Hausdorff Dimension of 2 (as demonstrated here - jstor.org). That's an integer - but still a problem because it's the "wrong" integer. (The standard dimension of a curve enclosing an area is one.). Moreover, I expect that given any complex (r,i) coordinate on that curve, a dr and di (or dx,dy if you prefer) can be determined. If that's true, then the degrees of freedom would seem to be solid equal to one.

But it is also interesting if the dr,di cannot be computed. It would a geometric example of something inherently undefined - reminiscent of HUP.