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\title{Final Exam \\ M 5500, Spring 2017}
\author{Solve any four problems}
\date{}
\maketitle

\begin{enumerate}
\item Find Euler equation and natural boundary condition, check Legendre condition
$$
I(u)=\min_{u(x)} \int_0^1\left((u')^2 + 3 \sin(x) u^2  \right) dx ~~\mbox{ if }
 \int_0^1\left(u  \right) dx=1
$$
\item Two equal masses $m=1$ with coordinates $r_1(t)$ and $r_2(t)$ move
in a plane, $(r_i(t)= (x_i(t), y_i(t))$ minimizing the difference of kinetic and potential energy
$$ V= \frac{\gamma}{| r_1 - r_2 |} $$ The initial conditions of the masses
are given.

Find  Lagrangian and Hamiltonian, equations of motion of the masses. Find
invariants to the motion. 
Hint: Express coordinates through the center of masses  $r_0=\frac{r_1+r_2}{2}$. 

\item Find an approximate solution to the problem
$$
I(u)=\min_{u(x)} \int_0^1\left(x(u'-1)^2 + 3 x^2 u^2  \right) dx ~u(0)=u(1)=0
$$
assuming a representation
$u(x)= (1-x) ( A_0+ A_1 x + A_2 x^2)$. Find the constants $A_0, A_1, A_2$.

\item Find Euler-Lagrange equation for the problem
$$
I(n)= \min_{u(x)} \left[ \int_\Omega \left((\nabla u)^2 - q(x) u \right) dx + \oint_{\partial
\Omega} \left(c(x)u+ d(x) u^2\right) ds \right],
$$
where $x=(x_1, x_2)$,  $u(x)$ is a scalar minimizer, $q(x), c(x), d(x) $
are given functions.

\item Minimize the length of the boundary between two domains of given
areas $A_1$ and $A_2$. Sketch the optimal shapes.


\end{enumerate}



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