3
$\begingroup$

I'm going through Zwiebach Chapter 6 on relativistic strings to try to solve a similar problem. I got all the way to my equation of motion

\begin{eqnarray*} \delta S & = & [ p' \delta \theta]_{z 0}^{z 1} + \int_{z 0}^{z 1} d z \left( p - \frac{\partial ( p')}{\partial z} \right) \delta \theta\\ & & \\ \Longrightarrow p - \frac{\partial ( p')}{\partial z} & = & 0\\ & & \\ {where} : & & \\ p & = & \frac{\partial L ( z, \theta, \theta')}{\partial \theta}\\ p' & = & \frac{\partial L ( z, \theta, \theta')}{\partial \theta'} \end{eqnarray*} The Lagrangian I have is $\begin{array}{lll} L ( z, \theta, \theta') & = & \frac{\cos^3 \theta}{z^5} \sqrt{1 + z^2 \theta'^2} \end{array}$, and I know that this equation is the Euler Lagrange equation.

\begin{eqnarray*} \frac{\partial L}{\partial \theta} - \frac{\partial}{\partial z} \left( \frac{\partial L}{\partial \theta'} \right) & = & 0\\ & & \\ {where} : & & \\ \theta' ( z) & = & \frac{\partial \theta ( z)}{\partial z}\\ L ( z, \theta, \theta') & = & \frac{\cos^3 \theta}{z^5} \sqrt{1 + z^2 \theta'^2} \end{eqnarray*} Just putting the expression for $L$ into the equation gives me a mess \begin{array}{lll} \frac{- 3 \sin \theta \cos^2 \theta}{z^5} \sqrt{1 + z^2 \theta'^2} - \frac{\sqrt{2} \theta'' ( 2 \theta'^2 z^2 + 3) \cos^3 \theta}{z^4 ( \theta'^2 z^2 + 2)^{3 / 2}} & = & 0 \end{array} I don't think I'm supposed to brute force this. I know the solution to be $\theta = \arcsin ( z)$, but I can't see how to get it. I'm thinking that there is something I'm missing about the equation of motion.

$\endgroup$

closed as off-topic by ACuriousMind, Rob Jeffries, JamalS, Kyle Kanos, Wolphram jonny Dec 16 '14 at 15:08

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, Rob Jeffries, JamalS, Wolphram jonny
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ oh and here is the link to the same question on physicsforums physicsforums.com/threads/… $\endgroup$ – user52205 Dec 16 '14 at 5:17
  • 1
    $\begingroup$ This question seems to have very little physical content; is it purely a question about the manipulations to solve your particular equation, or do you think there's something physical that's stopping you from getting it? $\endgroup$ – Danu Dec 16 '14 at 12:53
  • $\begingroup$ Which exercise in Zwiebach's book is this? Please provide context. $\endgroup$ – Qmechanic Dec 18 '14 at 23:02