$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ $\newcommand{\bfnu}{\boldsymbol{\nu}}$ ###[Problems to Chapter 10](id:sect-10.P) > 1. [Problem 1](#problem-10.P.1) > 2. [Problem 2](#problem-10.P.2) > 3. [Problem 3](#problem-10.P.3) > 4. [Problem 4](#problem-10.P.4) > 5. [Problem 5](#problem-10.P.5) ##[Problems](id:sect-10.P) There are several classical problems **[Problem 1.](id:problem-10.P.1)** The heavy flexible but unstretchable wire (chain) has a length and an energy respectively \begin{gather} L= \int\_0^a \sqrt{1+u^{\prime\,2}}\,dx, \label{eq-10.P.1}\\\\ U=\rho g \int\_0^a u \sqrt{1+u^{\prime\,2}}\,dx \label{eq-10.P.2} \end{gather} where $\rho$ is a linear density. a. Write down an equation minimizing energy $U$ as length $L$ is fixed. b. Find solution satisfying $u(0)=h\_0$, $u(a)=h\_1$. **[Problem 2.](id:problem-10.P.2)** We need to construct the fastest slide from point $(0,0)$ to $(a,-h)$. If $u(x)$ describes its shape then time is \begin{equation} T= \int\_0^a \frac{1}{\sqrt{2gu}} \sqrt{1+u^{\prime\,2}}\,dx. \label{eq-10.P.3} \end{equation} a. Write down an equation minimizing energy $U$ as length $L$ is fixed. b. Find solution satisfying $u(0)=0$, $u(a)=-h$. **[Problem 3.](id:problem-10.P.3)** If in 2D–light propagation the speed of light at point $(x,y)$ is $c(x,y)$ then the time of travel between two points $(x\_1,y\_1)$ and $(x\_2,y\_2)$ equals \begin{equation} T=\int\_{x\_1}^{x\_2} \frac{\sqrt{1+u'^2}\,dx}{c(x,u(x))} \label{eq-10.P.4} \end{equation} where $y=u(x)$ is equation of the ray, $u(x\_j)=y\_j$, $j=1,2$. a. Write down Euler' equation. b. Consider the case of $c(x,y)$ depending only on $y$ and reduce the 1-order equation to 1-st order one using conservation law $H(u,u')=\const$ where $H=u' L\_{u'}-L$ is a corresponding Hamiltonian, $L$ is a Lagrangian. **[Problem 4.](id:problem-10.P.4)** The area of the surface is \begin{equation} S=\iint\_{D} \sqrt{1+u\_x^2+u\_y^2}\,dxdy \label{eq-10.P.5} \end{equation} where $z=u(x,y)$, $(x,y)\in D$ is an equation of the surface. a. Write Euler-Lagrange PDE of the surface of the minimal area (with boundary conditions $u(x,y)=\phi(x,y)$ as $(x,y)\in \Gamma$ which is the boundary of $D$). b. if the potential energy is \begin{equation} E= kS - \iint\_D fu\,dxdy \label{eq-10.P.6} \end{equation} with $S$ defined by (\ref{eq-10.P.5}) and $f$ areal density of external force. Write Euler-Lagrange PDE of the surface of the minimal energy. **[Problem 5.](id:problem-10.P.5)** If the surface is a surface of revolution $z=u(r )$ with $r^2=x^2+y^2$ then \begin{equation} S=2\pi\int\_{D} \sqrt{1+u\_r^2}\,rdr \label{eq-10.P.7} \end{equation} Write Euler-Lagrange equation and solve it. --------------- [$\Leftarrow$](./S10.3.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](../Chapter11/S11.1.html)