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for this one, I can only do the first(a) $$V_{total}=\frac{mg}{l} \int_{x_0}^{d-x_0}{y(x)\sqrt{1+(\frac{dy(x)}{dx})^2}}dx$$

and have no idea about others.

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(a) Write down the potential energy of the rope as the function $y(x)$.

You're almost right, up to a minus sign in the limits of the integral: $$V=\dfrac{mg}{l}\int_{-x_0}^{d-x_0}y(x)\sqrt{1+y'^2}\ dx$$

(b) Since the problem is static, interpret the potential energy as the Lagrangian and find the Lagrangian density.

The Lagrangian is usually given by $L=T-V$. Since the problem is static, there is no kinetic energy and $T$ vanishes. Therefore: $$L=-V=-\dfrac{mg}{l}\int_{-x_0}^{d-x_0}y(x)\sqrt{1+y'^2}\ dx$$ Now, the Lagrangian density $\mathcal{L}$ is related to the Lagrangian $L$ by: $$L=\int\mathcal{L}\ dx$$ Such that: $$\mathcal{L}=-\dfrac{mg}{l}y\sqrt{1+y'^2}$$

(c) Using the Lagrangian density, construct the equation of motion.

The Euler-Lagrange equation for $\mathcal{L}(y,y',x)$ is: $$\dfrac{d}{dx}\dfrac{\partial\mathcal{L}}{\partial y'}-\dfrac{\partial\mathcal{L}}{\partial y}=0$$ Plugging in $\mathcal{L}$: $$\dfrac{d}{dx}\left[\dfrac{yy'}{\sqrt{1+y'^2}}\right]-\sqrt{1+y'^2}=0$$ which gives: $$\left[\dfrac{y'^2}{\sqrt{1+y'^2}}+\dfrac{yy''}{\sqrt{1+y'^2}}-\dfrac{y(y')^2y''}{(1+y'^2)^{3/2}}\right]-\sqrt{1+y'^2}=0$$ After some manipulations and simplifications: $$\dfrac{yy''}{(1+y'^2)^{3/2}}-\dfrac{1}{\sqrt{1+y'^2}}=0$$

(d) Solve for $y(x)$.

Multiplying through by $y'$ yields: $$\dfrac{yy'y''}{(1+y'^2)^{3/2}}-\dfrac{y'}{\sqrt{1+y'^2}}=\dfrac{d}{dx}\left[\dfrac{y}{\sqrt{1+y'^2}}\right]=0$$ Integrating with respect to $x$: $$\dfrac{y}{\sqrt{1+y'^2}}=C_1,\quad C_1\in\mathbb{R}$$ which you may recognize as the first-order differential equation for a catenary, such that we get: $$y=C_1\cosh\left(\dfrac{x+C_2}{C_1}\right),\quad C_2\in\mathbb{R}$$ All that is left to do is to find $C_1$ and $C_2$ from the boundary conditions on $y(x)$.

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  • $\begingroup$ thanks so much... do you have any suggestion book for Analytical mechanics for new learners?.. I am using the David Tong's <Classical Dynamics> lecture note which has few about lagrangian density... $\endgroup$
    – leave2014
    Mar 21, 2015 at 13:23
  • $\begingroup$ You're welcome! David Tong's lecture notes are very good, but I would also suggest taking a look at Mathematics for Physics by Stone and Goldbart. This book covers a wide range of topics in maths and physics, very useful to prepare for graduate level physics. The first part should cover everything you need to know about Lagrangians, Lagrangian densities, calculus of variation, etc. (You might be able to find chapters in PDF format, since the full book is a bit expensive - but worth it!) $\endgroup$
    – Demosthene
    Mar 21, 2015 at 13:30

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