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\title{Final Exam \\ M 5500, Spring 2017}
\author{Andrej Cherkaev}
\date{}
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\begin{enumerate}
\item Find Euler equation and boundary condition, check Legendre condition
$$
I(u)=\min_{u(x)} \int_0^1\left((u')^2 + 3 sin(x) u  \right) dx - u(1)
$$
\item Two equal masses $m=1$ with coordinates $r_1(t)$ and $r_2(t)$ move in a plane 
minimizing the potential energy 
$$ f= \frac{\gamma}{{|r_1 = r_2|} $$ The initial conditions of the masses are given.

Find  Lagrangian and Hamiltonian, equations of motion of the masses. Find invariants to the motion. 

\item The input signal $ v(x)= \arctan (.01 x)$ contains  a noice $\eta(x) =.2 \cos (13 x)+cos(10 x )$ To denoice the signal, we solve the problem
$$
I(u)=\min_{u(x)} \int_0^{2\pi}\left( (v+ \eta -u)^2+ c^2 (u')^2  \right) dx 
$$
Derive the Euler equation for  $u$ and natural boundary conditions.



Filter it, plot the results
\item Optimal shape
\item Find approximate solution
\end{enumerate}



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