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\title{Final Exam \\ M 5500, Spring 2018}
\author{Solve any six problems}
\date{Return at Monday, April 30, before 5 pm}
\maketitle

\begin{enumerate}
\item Find Euler equation and natural boundary condition, check Legendre condition for the isoperimetric problem
$$
I(u)=\min_{u(x)} \int_0^1\left((u')^2 - 3 u\,u'+ 3 \sin(x) u^2  \right) dx ~~\mbox{ if }
 \int_0^1\left(u  \right) dx=1
$$
\item Two equal  gravitating masses $m=1$ with coordinates $r_1(t)$ and $r_2(t)$ move
in a plane, $$r_i(t)= \left(x_i(t), y_i(t)\right)$$  Gravitational force is: 
$$ F= \gamma \frac{r_1-r_2}{| r_1 - r_2 |^3} $$ 
Derive kinetic and potential energy, formulate a variational problem.
Find  Lagrangian, Hamiltonian, and equations of motion of the masses. Find
invariants to the movement. 

{\it Hint: } Express coordinates through the center of masses  $r_0=\frac{1}{2}(r_1+r_2)$ and deviation from the center. 

\item Consider the problem
$$
I= \min_{u(x)} \int_0^1 [a \,(u')^2 - c\, u(u-1)] dx, \quad u(0)=0
$$
a. Write Hamiltonian and dual Lagrangian.
\newline
b. Derive Euler equations and boundary conditions for the primary and the dual problems, derive system of first-order equation and boundary conditions. 

\item A process is described by a differential equation
$$
- a u'' + c u^3+ 1=0, ~\mbox{ in } (0, 1) \quad u(0)=0, ~ u(1) + u'(1)=3
$$
Write a variational problem (Lagrangian and the boundary term) for which this equation is the Euler equation. 

\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
$u(x)=x  (1-x) ( A_0+ A_1 x + A_2 x^2)$. Find the constants $A_0, A_1, A_2$. You may use Maple or similar software. 

\item Show that the Euler equation of the problem 
$$ 
\mbox{inf}_{u(x)} \int_0^1 \left( (u-x)^2 + \min\left\{(u'-1)^2,~ (u'+1)^2 \}\right\} \right)dx, \quad u(0)=0 
$$
does not satisfy the Weierstass test. 
 Describe a minimizing sequence. Find the relaxed formulation of the problem. 

\item Find Euler-Lagrange equation and natural boundary conditions for the problem
$$
I(n)= \min_{u(x)} \left[ \int_\Omega \left((\nabla u)^2 - q\, u \right) dx + \oint_{\Gamma} \left((c\,u+ d\, u^2\right) ds \right], ~\int_\Omega u^2 \, dx=1
$$
where $x=(x_1, x_2)$,  $\Omega$ is a bounded domain in $R_2$ with the smooth boundary $\Gamma$, $u(x)$ is a scalar minimizer, $q, ~c,  ~ d$ are parameters.

\end{enumerate}

\vspace{0.15in}
{\Large \bf Extra-Credit Problem}

\vspace{0.15in}
Show that the Euler equation of the problem 
$$ 
\mbox{inf}_{u(x)} \int_0^1 \left( (u-\alpha x)^2 + \min\left\{(u'-1)^2,~ (u'+1)^2 \}\right\} \right)dx, \quad u(0)=0 
$$
does not satisfy the Weierstass test for certain values of $\alpha$. Find these values. 
 Describe a minimizing sequence. Find the relaxed formulation of the problem. 







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