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\begin{document}

\title{Differential Equations}
\author{Homework 6}
\date{Due in class Tuesday, April 17, 2018}
\maketitle

\begin{enumerate}


\item (Verhulst, Exercise 10.2.) Consider the mathematical pendulum 
\[
  \ddot x + \sin x = 0 
\]
with initial conditions $x(0) = a$, $\dot x(0) = 0$.
\begin{enumerate}
\item Setting $x = a \, y$, show that the system is equivalent to 
\[
  \ddot y + y = \frac{a^2}6 \, y^3 + O(a^4) \,.
\]
Thus, argue that the period of oscillations $T$ is slowly changing
with $a$.

\item Use the Poincar\'e--Lindstedt method to show find
\[
  T(a) = 2 \pi \, 
  \biggl( 1 + \frac{a^2}{16} + O(a^4) \biggr) \,.
\]
\emph{Note}: This problem is degenerate, i.e., the perturbation does
not select a specific periodic orbit.  Yet, you can write out the
periodicity conditions, expand in powers of $a$, and solve them.
\end{enumerate}

\item  (Verhulst, Exercise 11.4.)
A satellite moves in the outer atmosphere of a spherically symmetric,
homogeneous planet.  The (nondimensionalized) equations of motion read
\[
  \ddot \r = - \frac{\r}{r^3} - \varepsilon \, \dot \r 
\]
where $\r = (x,y,z)$ and $r = \lVert \r \rVert$.
\begin{enumerate}
\item Show that for given initial conditions, the motion takes place
in a plane through the origin (in physical space, not phase space!).
Thus, in the following, you can take $z=0$ without loss of generality.

\item Introduce polar coordinates $x = r \, \cos \theta$ and $y = r \,
\sin \theta$ to find
\begin{gather*}
  \ddot r - r \, \dot \theta^2 
  = -\frac1{r^2} - \varepsilon \, \dot r \,, \\
  2 \, \dot r \, \dot \theta + r \, \ddot \theta 
  = - \varepsilon \, r \, \dot \theta \,.
\end{gather*}
\item Integrate the second equation to find
\[
  r^2 \, \dot \theta = c \, \e^{-\varepsilon t} \,.
\]
(The constant $c$ is the initial angular momentum of the satellite.)

\item Set $\rho = 1/r$ and use $\theta$ as a time-like variable.  Show
that this results in
\begin{gather*}
  \frac{\d^2 \rho}{\d \theta^2} + \rho = u \,, \\
  \frac{\d u}{\d \theta} = 2 \, \varepsilon \, \frac{u^{3/2}}{\rho^2} 
\end{gather*}
with 
\[
  u = \frac1{c^2} \, \e^{2\varepsilon t} \,.
\]

\item Obtain a Lagrange standard form by writing
\begin{gather*}
  \rho = u + a(\theta) \, \cos \theta + b(\theta) \, \sin \theta \,,
  \\
  \frac{\d \rho}{\d \theta} = - a(\theta) \, \sin \theta + b(\theta)
  \, \cos \theta \,.
\end{gather*}
\item Apply averaging and give the approximations for $\rho(\theta)$
and $r(t)$ with an initially circular orbit where $\theta(0) = 0$,
$r(0) = c^2$, and $\dot r(0) = 0$.  Discuss the asymptotic validity of
the approximation.
\end{enumerate}


\end{enumerate}





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