$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ $\newcommand{\bfnu}{\boldsymbol{\nu}}$

There are several classical problems.

**Problem 1.**
We need to construct the fastest slide from point $(0,0)$ to
$(a,-h)$. If $y(x)$ describes its shape then time is
\begin{equation}
T= \int_0^a \frac{1}{\sqrt{2guy}} \sqrt{1+y^{\prime\,2}}\,dx.
\label{eq-10.2.P.1}
\end{equation}
We need to minimize $T$.

- Write Euler-Lagrange equation and integrate it.
- Find solution satisfying $y(0)=0$, $y(a)=-h$ (so that the horizontal length and the hight of the slide are given).
- Write down boundary condition, if $y(a)$ is not given (so that only the horizontal length of the slide is given). Find corresponding solution.

The resulting curve is called*brachistochrone*.

**Problem 2.**
The heavy flexible but unstretchable wire (chain) has a length and an energy respectively
\begin{gather}
L= \int_0^a \sqrt{1+y^{\prime\,2}}\,dx,
\label{eq-10.2.P.2}\\
U=\rho g \int_0^a y \sqrt{1+y^{\prime\,2}}\,dx
\label{eq-10.2.P.3}
\end{gather}
where $\rho$ is a linear density.
We need to minimize energy energy $U$ as length $L$ is fixed.

- Write Euler-Lagrange equation and integrate it.
- Find solution satisfying $y(0)=h_0$, $y(a)=h_1$.

The resulting curve is called*catenary*.

**Problem 3.**
Consider curve $y=y(x)$. The length of it and the area bounded by it are respectively
\begin{gather}
L=\int_0^a \sqrt{1+y^{\prime\,2}}\,dx,\\
A=\int_0^a ydx.
\end{gather}
We need to maximize $A$ as length $L$ is fixed.

Write Euler-Lagrange equation and integrate it.

**Problem 4.**
Consider curve $y=y(x)$. The area of the surface of the revolution formed by it and the volume inside of it are respectively
\begin{gather}
S=2\pi\int_0^a y\sqrt{1+y^{\prime\,2}}\,dx,\\
V=\pi\int_0^a y^2\,dx.
\end{gather}

- We need to minimize $S$ as $y(0)=b$, $y(a)=c$ are fixed.

Write Euler-Lagrange equation and integrate it. - We need to maximize volume $V$, as ares $S$ is given and
$y(0)=y(a)=0$.

Write Euler-Lagrange equation and integrate it.

**Problem 5.**
If in 2Dâ€“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+y'^2}\,dx}{c(x,y(x))}
\end{equation}
where $y=y(x)$ is equation of the ray, $u(x_j)=y_j$, $j=1,2$.

- Write Euler-Lagrange equation.
- Integrate it if $c=c(y)$.

**Problem 6.**
We need to minimize
\begin{equation}
\int_0^a (u''^2 -2 u)\,dx
\end{equation}
under constrains below.

- Write Euler-Lagrange equation.
- Find solution, satisfying $u(0)=u'(0)=u(a)=u'(a)=0$.
- If only $u(0)=u'(0)=u(a)=0$ are given, write a second condition at $x=a$ and find a corresponding solution.
- If only $u(0)=u'(0)=u'(a)=0$ are given, write a second condition at $x=a$ and find a corresponding solution.
- If only $u(0)=u'(0)=0$ are given, write two conditions at $x=a$ and find a corresponding solution.

**Problem 7.**
We need to minimize
\begin{equation}
\int_0^a u''^2 \,dx
\end{equation}
under constrain
\begin{equation}
\int_0^a u^2 \,dx=1
\end{equation}
and constrains below.

- Write Euler-Lagrange equation.
- Find solution, satisfying $u(0)=u'(0)=u(a)=u'(a)=0$.
- If only $u(0)=u'(0)=u(a)=0$ are given, write a second condition at $x=a$ and find a corresponding solution.
- If only $u(0)=u'(0)=u'(a)=0$ are given, write a second condition at $x=a$ and find a corresponding solution.
- If only $u(0)=u'(0)=0$ are given, write two conditions at $x=a$ and find a corresponding solution.