Extremising a functional with boundary conditions (Euler-Lagrange)

Question

I need to determine all functions $ u(x) $ that extremise the functional: $$ I[u]= \int_{-\infty}^\infty \left[\frac{(u')^2}{2}+(1-\cos u)\right] \, dx $$ subject to the boundary conditions

$$ \lim_{x \to -\infty} u(x)=0 $$ and $$ \lim_{x \to \infty} u(x) = 2\pi $$ I used the standard approach for finding the stationary points of a functional; that is, attempting to solve the Euler-Lagrange equation, but assuming I've attempted this correctly I arrive at $$ \frac{d^2u}{dx^2} = \sin u $$ which I believe is not (easily) directly solvable, so I'm presuming there's either another way to approach this problem or I've messed up somewhere. A point in the right direction would be great, thanks in advance

Answer 1

$\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$

\begin{equation} \lim_{x \to -\infty}\,\mrm{u}\pars{x} = 0\,,\qquad \lim_{x \to \infty}\,\mrm{u}\pars{x} = 2\pi\label{1}\tag{1} \end{equation}

Multiply both sides of your differential equation $\ds{\totald[2]{\mrm{u}}{x} = \sin\pars{\mrm{u}}}$ by $\ds{\totald{\mrm{u}}{x}}$: \begin{align} &\totald{\mrm{u}}{x}\,\totald[2]{\mrm{u}}{x} = \totald{\mrm{u}}{x}\,\sin\pars{\mrm{u}} \implies {1 \over 2}\bracks{\totald{\mrm{u}}{x}}^{2} = -\cos\pars{\mrm{u}} + \mc{E} + 1\ \mbox{where}\ \,\mc{E}\ \mbox{is a}\ constant. \end{align}

---

Also, \begin{align} {1 \over 2}\bracks{\totald{\mrm{u}}{x}}^{2} & = 2\sin^{2}{u \over 2} + \,\mc{E} \implies \totald{u}{x} = \pm\root{2\,\mc{E} + 2\sin^{2}\pars{u \over 2}} \\[5mm] \implies \pm\int{\dd u \over \root{2\,\mc{E} + 2\sin^{2}\pars{u/2}}} &= x + \,\mc{C}\,,\quad\mc{C}\ \mbox{is a}\ constant. \end{align} Note that the integral is related to the Elliptic Integral $\,\mrm{F}\pars{\phi,k}$. Now, you have to manage to find $\ds{\,\mc{E},\ \mc{C}}$ and the 'suitable' sign $\ds{\pm}$ by using the boundary conditions \eqref{1}.