\documentclass[12pt]{article}
\pagestyle{empty}
\usepackage{amsfonts,amsmath}

\def\Rl{\mathbb{R}}
\def\Cx{\mathbb{C}}

\begin{document}

\centerline{{\sc Problem set 2}}
\centerline{Math 207B, Winter 2016}
\centerline{Due: Fri, Jan. 29}


\bigskip\noindent
{\bf 1.} A particle of mass $m$ with position $\vec{x}(t)$ at time $t$ has potential energy $V(\vec{x})$ and kinetic energy
\[
T = \frac{1}{2} m |\vec{x}_t|^2.
\]
The action of the particle over times $t_0 \le t \le t_1$ is the time-integral of the difference between the kinetic and potential energy:
\[
S(\vec{x}) = \int_{t_0}^{t_1} \left(T-V\right)\, dt.
\]
(a) Show that an extremal $\vec{x}(t)$ of $S$ satisfies Newton's second law $\vec{F}  = m\vec{a}$ for motion in a conservative force field
$\vec{F} = - \nabla V$.

\smallskip\noindent
(b) Show that the total energy of the particle $E = T +V$ is a constant independent of time.



\bigskip\noindent
{\bf 2.} Let $\Omega \subset \Rl^n$ be a bounded region with smooth boundary (so the divergence theorem holds) and $f : \overline{\Omega} \to \Rl$ a smooth function. Derive the Euler-Lagrange equation and natural boundary condition that are satisfied by a smooth extremal $u : \overline{\Omega} \to \Rl$ of the functional
\[
J(u) = \int_{\Omega} \left(\frac{1}{2}|\nabla u|^2 - f u\right)\, dx.
\]

\bigskip\noindent
{\bf 3.} The transverse displacement at position $x$ and time $t$ of an elastic string vibrating in the $(x,y)$-plane is given by $y = u(x,t)$, where $a\le x \le b$ and $t_0 \le t \le t_1$. If the density of the string per unit length is $\rho(x)$ and the tension in the string is a constant $T$, then (for small displacements) the motion of the string is an extremum of the action
\[
S(u) = \int_{t_0}^{t_1} \int_a^b \left(\frac{1}{2} \rho u_t^2 - \frac{1}{2} T u_x^2\right)\, dx dt.
\]
Derive the Euler-Lagrange equation for $u(x,t)$.

\bigskip\noindent
{\bf 4.} The ($n$-dimensional) area of a surface $y = u(x)$ over a region $\Omega \subset \Rl^n$ is given by
\[
J(u) = \int_{\Omega} \sqrt{1 + |\nabla u|^2}\, dx.
\]
Find the Euler-Lagrange equation (called the minimal surface equation) that is satisfied by a smooth extremum of this functional.


\bigskip\noindent
{\bf 5.} Let $
X = \left\{u\in C^1([-1,1]) : u(-1) = -1, u(1) = 1\right\},
$
where $C^1([a,b])$ denotes the space of continuously differentiable functions on $[a,b]$. Define $J : X \to \Rl$ by
\[
J(u) = \int_{-1}^1 x^4 (u')^2\, dx.
\]
(a) Show that
\[
\inf_{u\in X} J(u) = 0,
\]
but $J(u) > 0$ for every $u\in X$ (so $J$ does not attain its infimum on $X$).

\smallskip\noindent
(b) What happens when you try to solve the Euler-Lagrange equation for extremals of $J$?  


\end{document}             % End of document.
