Problems to Sections 10.1, 10.2


### Problems to Sections 10.1, 10.2

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 $$T= \int_0^a \frac{1}{\sqrt{2guy}} \sqrt{1+y^{\prime\,2}}\,dx. \label{eq-10.2.P.1}$$ We need to minimize $T$.

1. Write Euler-Lagrange equation and integrate it.
2. Find solution satisfying $y(0)=0$, $y(a)=-h$ (so that the horizontal length and the hight of the slide are given).
3. 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.

1. Write Euler-Lagrange equation and integrate it.
2. 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}

1. We need to minimize $S$ as $y(0)=b$, $y(a)=c$ are fixed.
Write Euler-Lagrange equation and integrate it.
2. 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 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 $$T=\int_{x_1}^{x_2} \frac{\sqrt{1+y'^2}\,dx}{c(x,y(x))}$$ where $y=y(x)$ is equation of the ray, $u(x_j)=y_j$, $j=1,2$.

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

Problem 6. We need to minimize $$\int_0^a (u''^2 -2 u)\,dx$$ under constrains below.

1. Write Euler-Lagrange equation.
2. Find solution, satisfying $u(0)=u'(0)=u(a)=u'(a)=0$.
3. If only $u(0)=u'(0)=u(a)=0$ are given, write a second condition at $x=a$ and find a corresponding solution.
4. If only $u(0)=u'(0)=u'(a)=0$ are given, write a second condition at $x=a$ and find a corresponding solution.
5. 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 $$\int_0^a u''^2 \,dx$$ under constrain $$\int_0^a u^2 \,dx=1$$ and constrains below.

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