# Differential Geometric Aspects of Rubber Bands

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?
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I assume you intend that the stretching should be uniform along the entire length of the rubber band. In particular, you don't want that, for example, half of the rubber band remains unstretched while the rest stretches to cover the desired curve. – Andreas Blass Aug 30 '14 at 13:11
@AndreasBlass yes, your assumption is right, the force due to elongation is the same in each point. – Manfred Weis Aug 30 '14 at 13:28
I think there may be a contradiction in your question: you assume that $l_0 = 0$ and that the force obeys Hooke's law, i.e. that it scales linearly with the strain relavive to the rest state. That would mean that the force is infinite for any $l_1 > 0$. Also, I think the band would not contract to a single point due to energy preservation: it would overshoot and keep oscillating like a spring does. – Jaap Eldering Aug 30 '14 at 14:34
@JaapEldering I wrote that the initial length shall be infinitesimally small, which isn't the same as 0 (at least to my understanding); concerning your other point: I agree, that I didn't address all peculiarities of physical reality; but would they be of any help here? – Manfred Weis Aug 30 '14 at 15:20
What do you mean by infinitesimally small then, if not the same as 0? Please give a mathematical definition. – Hans Aug 31 '14 at 1:17

Let me give a shot at a partial answer, and provide a model of the problem that I hope is physically sensible. First, since we're actually dealing with a simplification of elasticity theory, and Hooke's Law can be viewed as a specific instance of linearized elasticity, this means that the stored energy function (i.e. potential energy density) is quadratic in strains away from the rest state of the rubber band. Let me here assume that the band has rest length $l_0 = 2\pi$ and that it is parametrized by $\phi\colon S^1 \to \mathbb{R}^3$. Then the stored energy is reasonably given by $$W(s) = \frac{k}{2}(\|\phi'(s)\| - 1)^2.$$ Assuming a uniform mass density $\rho$, we have a Lagrangian $$L(\phi,\dot{\phi}) = \int_0^{2\pi} \frac{\rho}{2}\|\dot{\phi}(s)\|^2 - \frac{k}{2}(\|\phi'(s)\| - 1)^2 \;ds.$$ Since $L$ is invariant under translations (and rotations), the total momentum $$P = \rho \int_0^{2\pi} \dot{\phi}(s) \;ds,$$ is a conserved quantity. Since $P = 0$ initially, and $P/\rho$ is the change of the center of mass, it follows that the center of mass is constant in time. In particular, if at any time the rubber band contracts to length zero, it will be at its center of mass point. This answers your second question. (Note that this derivation only depended on $L$ being invariant under translations, not the specific stored energy function.)
Yes. He should just let $l_0>0$ and ask, given the initial curve and, say, assuming the initial tension is all the same throughout the band, what the trajectory of the point on the rubber band is. – Hans Aug 31 '14 at 1:20