What happens, if a rubber band ( of length $l_0$ that has been stretched to length $l_1:=l_0+\Delta l\;$ and brought into the shape of a closed curve in $\mathbb{R}^3$ ) is released and if the only force at work is due to the elongation and obeys Hooke's law?

Clarification in response to Andreas' comment:

The force due to stretching shall be the same in each point of the rubberband.

Clarification in response to Hansen's comments:

for the mathematical discussion of the problem, it shall be assumed that $l_0=0$, so that the contraction doesn't stop as long as the rubber band has positive length.

Furthermore, I would like the mathematical question to be discussed on basis of points and vectors of $\mathbb{R}^3$; physical phenomena like moment of inertia, or bending energy, etc., shall not play a role in this context.

I acknowledge however, that trying to model real-world rubber bands is also an interesting question to be tackled, after the questions related to the (over-)simplified model have been solved.

By "what happens", I mean

- what kind of surface is traced out by an infinitely thin rubber band of infinitesimal small initial length $l_0$?
- what are the coordinates of the point, to which the rubber band contracts, as its length tends to 0?
- what are the trajectories of the points on the rubber band during contraction?