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I've been thinking about creating a continuous rope, made out of infinitely many springs, with infinitely small distances between them.

At first, I developed the Euler-Lagrange Equations: $$ \mathcal{L} = T - U \\ T = \frac{1}{2}m \dot{r}^2 \\ U = \frac{1}{2}kr^2 \\ \mathcal{L} = \frac{1}{2}m \dot{r}^2 - \frac{1}{2}kr^2 $$ But because there are $n$ springs, what I will have to do is to sum up all the spring velocities and distances: $$ \mathcal{L} = \sum_{i=0}^{n} \left [ \frac{1}{2}m \dot{r_i}^2 - \frac{1}{2}k(r_{i+1} - r_i)^2 \right ] $$ Taking the limit to the infinity: $$ \mathcal{L} = \lim_{n \to \infty} \sum_{i=0}^{n} \left [ \frac{1}{2}m \dot{r_i}^2 - \frac{1}{2}k(r_{i+1} - r_i)^2 \right ] = \int \left [ \frac{1}{2}m \dot{r_i}^2 - \frac{1}{2}k(r_{i+1} - r_i)^2 \right ] $$ Rearanging a little bit the equation, we finally get: $$ \mathcal{L} = \frac{1}{2} m \int_{- \infty}^{\infty} \dot{r}^2 - \frac{1}{2} k \int_{- \infty}^{\infty} (r_{i+1} - r_i)^2 $$ But here, I'm stuck. I've tried to parametrize the integral and find any antiderivative, but I don't have very clear how to do it exactly. Is there anything I've being doing wrong? Any advice? Is this even possible?

Álvaro Rodrigo
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  • So your final system's mass is infinite, as is its spring constant? I would assume that both these quantities would need to change appropriately if you wish to have a sensible result. – Philip Aug 26 '21 at 18:13
  • I'm assuming that the masses joined to the springs all have the same mass, and is constant? Is this a mistake? How would i solve it? – Álvaro Rodrigo Aug 26 '21 at 18:28
  • Well, what it seems to me you're trying to do is solve for a continuous system which will finally have a finite mass. As a result, you should be using the mass density to define the problem. That way, when you let $n \to \infty$, there will be many more tiny masses, but the total mass of the system remains constant. A similar argument can be used for $k$, except (I suspect) each $k$ will be proportional to $n$. This is an interesting though slightly uncommon approach. The end result for the dynamics will, of course, just be the wave equation, though I've never calculated $\mathcal{L}$... – Philip Aug 26 '21 at 18:40
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    Related: https://physics.stackexchange.com/q/133664 – AlmostClueless Aug 26 '21 at 18:45
  • Keith's answer in the above linked question is quite accessible. There is also this set of notes which deals with the subject as you want to, but the mathematics is relatively involved. This is a nice introductory problem to acquaint someone with field theory... – Philip Aug 26 '21 at 18:51

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