Write down the Euler-Lagrange equation for the functional

$\displaystyle \displaystyle{I[u]=\int_{-\infty}^{\infty}\left[\frac{1}{2}\,u'^{2}+(1-\cos(u))\right]dx}$

and find all solutions which satisfy $\displaystyle \lim_{x\to-\infty}u(x)=0$ and $\displaystyle \lim_{x\to+\infty}u(x)=2\pi.$ Show that if $\displaystyle u\in C^{1}(\mathbb{R})$ satisfies $\displaystyle \lim_{x\to-\infty}u(x)=0$ and $\displaystyle \lim_{x\to+\infty}u(x)=2\pi$

$\displaystyle \displaystyle{I[u]=\frac{1}{2}\int_{-\infty}^{\infty}\left(u'-2\sin\left(\frac{u}{2}\right)\right)^{2}dx+8.}$

Deduce that a lower bound for $\displaystyle I[u]$ amongst such functions is $\displaystyle 8$, and give a

**first order** differential equation which $\displaystyle u$ must satisfy in order to realize this lower bound. Show that any solution of this first order equation solves the Euler-Lagrange equation you derived in the first part of the question. Give all the functions satisfying $\displaystyle I[u]=8.$