\documentclass[12pt]{article}
\usepackage{epsfig}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}

{\bf Question}

A simple pendulum has angular position $\theta$ and angular
momentum $p$. The motion of the pendulum (assuming a suitable set
of measurement units so there are no constants in the equation)
can then be described by the following ordinary differential
equation for $p(\theta)$ $$ \frac{dp}{d\theta}= -\frac{\sin
\theta}{p}\;. $$ Sketch the direction field (note the periodicity
and show values of $-2\pi\le\theta\le2\pi$). Comment on the
different behaviour between a solution that has a very small value
of $p$ when $\theta=0$ and a solution that has very large $p$ when
$\theta=0$.$\qquad (*)$






\vspace{0.25in}

{\bf Answer}

\textbf{Isoclines} are $c=-\frac{\sin\theta}{P}$.
$$\Rightarrow -cP = \sin\theta$$

\begin{center}
$\begin{array}{c}
\epsfig{file=114-1-10.eps, width=60mm}
\end{array}
\ \ \
\begin{array}{cc}
c=0 & \theta = n\pi\\
& n=0,1,2\cdots\\
c=1 & P =-\sin\theta\\
c=2 & P = \sin\theta\\
c\to\infty & P=0
\end{array}$
\end{center}

\begin{center}
\epsfig{file=114-1-11.eps, width=70mm}
\end{center}

If $P$ is small when $\theta=0$ then the solution $P(\theta)$ only
exists for values of $\theta$, $-\pi < -\theta_0 \le \theta \le
\theta_0 < \pi$. e.g.

\begin{center}
\epsfig{file=114-1-12.eps, width=50mm}
\end{center}
(A pendulum oscillating back and forth)

If $P$ is large when $\theta =0$ then the solution $P(\theta)$ exists
for all $\theta$. e.g.

\begin{center}
\epsfig{file=114-1-13.eps, width=50mm}
\end{center}
(A pendulum swinging over and over).


\end{document}
