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I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points.

First of all I wrote down this Lagrangian:

$$S=\int dt ds \left[\frac{\rho}{2}(\dot{x}^2+\dot{y}^2)-\rho g y(s,t)+\frac{\lambda(s,t)}{2}\left(\left(\frac{\partial x}{\partial s}\right)^2+\left(\frac{\partial y}{\partial s}\right)^2-1\right)\right]$$

This Lagrangian describes heavy string with fixed ends in gravitational field. Where $\rho$ is density, $g$ is gravitational acceleration, $s$ is natural parameter.

So I have 3 equations from Euler-Lagrange equations.

$$\rho\ddot{x}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial x}{\partial s }\right)=0$$ $$\rho\ddot{y}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial y}{\partial s }\right)+\rho g=0$$ $$\left(\frac{\partial x}{\partial s }\right)^2+\left(\frac{\partial y}{\partial s }\right)^2=1$$

After that I've found stationary solution ($\frac{\partial x}{\partial t}=\frac{\partial y}{\partial t}=\frac{\partial \lambda}{\partial t}=0$). (I just put $\ddot{x}=\ddot{y}=0$)

$$y_0(x)=-\frac{C_1}{\rho g}\cosh\left(\frac{\rho g x}{C_1}+C_2\right)$$

Where $C_1,C_2$ is integration constants (depends on positions of ends of string). And $\cosh(x)$ is hyperbolic cosine.

To study small oscillations I've tried to use pertrubation theory.

So, I put $$y(s,t)=y_0(s)+\bar{y}(s,t)$$ $$x(s,t)=x_0(s)+\bar{x}(s,t)$$ $$\lambda(s,t)=\lambda_0(s)+\bar{\lambda}(s,t)$$

But after that I get difficult differential equations, which I can't solve.

Maybe someone know the more simplier aproach to solve this problem or know how to solve it in this way?

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    $\begingroup$ I did the same thing (7-8 years ago) to find the galloping modes of an overhead power line (catenary + vibration), but I do not remember how I did it. If I come up with something I will try to post. $\endgroup$ – ja72 Oct 14 '13 at 19:47
  • $\begingroup$ @ja72 It would be great! :) $\endgroup$ – Oiale Oct 15 '13 at 15:03
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    $\begingroup$ I don't know if this will help, but I would parametrize the perturbation by a movement perpendicular to the static string. The second thing I would look into is if the movement could be treated as a wave in a media with a varying speed of travel. I base this on the idea that the tension is greater near the ends and so the speed of the wave is greater there as well. $\endgroup$ – John M. Cavallo Oct 16 '13 at 0:00
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    $\begingroup$ I guess there are longitudonal and transverse waves. Is your focus on both, or one of them? $\endgroup$ – ja72 Oct 16 '13 at 15:10
  • $\begingroup$ @ja72 Now I'm focusing on transverse waves. $\endgroup$ – Oiale Oct 16 '13 at 17:18
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Here is a link that explains how to do it. You need to expand the Lagrangian around the steady solution. That should give you an easier set of differential equations for the small perturbation. Hope this helps.

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I was also involved in this problem for the past few weeks.You can write newtons law of motion for small segment of string and obtain a differential equation.From that equation you can find the normal modes of the string and the general motion of the string is given by a superposition of the normal modes.But i ignored the longitudinal oscillations and considered only the transverse motion.The solution was in terms of Bessel function of zeroth order.

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