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\begin{document}

\begin{center}
{\Large \em Math 560-61 (1998-99)}\\
\vspace{.3in}
{\Large Assignment 7}\\
\end{center}

\vspace{.4in}

\large
\begin{itemize}
\item[1.] Let $R$ be some bounded smooth domain in $\reals^2$
and $\partial R$ its bounding curve and define
\eq
J(u) = \int \int_R F(x,y,u,u_x,u_y) dA + \int_{\partial{R}}
\vec{G} \cdot \vec{dr}
\endeq
where $\vec{dr}=(dx,dy)$ and 
$\vec{G} = ( G(x,y,u),H(x,y,u))$. Use Green's Theorem in the 
plane to deduce the Euler Lagrange equations and natural boundary
conditions for the above functional.

If the Euler-Lagrange equations for the problem turn out to  be 
\eq
u_x u_{xx} + u_y u_{yy} = 0 \quad , u=u(x,y)
\endeq
what is $F$?
(Note: The answers are not unique and may require assuming
$F_u=0$ and/or $F_x=F_y=0$)

\item[2.] Let $R$ be the unit sphere in $\reals^3$
and $\partial R$ its bounding surface. Let 
\eq
J(u) = \int \int \int_R x^2 u_x^2 +y^2 u_y^2 +z^2 u_z^2 dV
\endeq
and
\[
{\cal{A}}= \{u \in C^2(\bar{R}): u|_{\partial R}= 0, 
\parallel u \parallel = 1 \}
\]
where the norm is the $L^2(R)$ sense.
If $\bar{u}$ minimizes $J$ on ${\cal{A}}$, what eigenvalue
problem does $\bar{u}$ solve? Use the variational formulation
to find any bound for the smallest (positive) eigenvalue.

\item[3.] A mass $m$ is suspended from a pivot
located at $(x,y,z)=(0,0,a)$, $a>0$ with an elastic rigid piston
so that it is always in contact with a parabolic surface
$z=x^2+y^2$. Assume the piston has neglible mass, 
the spring constant of the piston is $k$, gravity
is constant and that the motion is frictionless.
Using cylindrical coordinates $(r,\theta,z)$, derive the Lagrangian,
conugate momenta, the Hamiltonian, Lagrange's equations
for motion and Hamilton's equations for the motion.
Reduce one or both of these sets of equations to
a single differential equation for $r(t)$.

\end{itemize}

\end{document}