I was trying to derive the function $h(x)$ that describes the shape of a string fixed at two points with distance $D$ (which aligned with the x-axis), in other words, $h(0)=h(D)=0$ but I got stuck.

I am interested in a solution that involves Lagrangian mechanics, as I am curious to know how one should solve a problem where the object in question is not rigid with the tool. All I could get is that the integral $$\int_0^D\sqrt{1+h'(x)}\mathrm dx=L,$$ where $L$ is the length of the string.

  • $\begingroup$ A Fourier series of sine waves? $\endgroup$ – Pieter Dec 30 '19 at 20:00
  • $\begingroup$ This might help: math.stackexchange.com/q/2261921. The shape you are describing is called catenary. $\endgroup$ – Mr. Xcoder Dec 30 '19 at 20:07