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\noindent
Math 555 \hfill Winter 2015\\
{\bf Homework 3} \hfill {\bf Due Friday, January 23}    

%\bigskip\medskip
%\noindent
%\begin{tabular}{lp{4.8in}}
%\kern-.1in
%{\bf Reading:} & Coddington \& Levinson, Chapter 1 and begin Chapter 3
%\end{tabular}

\bigskip

\begin{enumerate}
\item
Let $A=\left[\begin{array}{cc}10& 6\\ -18&-11\end{array}\right]$

\medskip
\begin{enumerate}
\item
Compute $e^{tA}$.
\item
Find the general solution of $x'=Ax$.
\end{enumerate}

%\bigskip
%\item
%Let $A\in\C^{n\times n}$ and let $\lambda \in \C$ be an eigenvalue of $A$
%of algebraic multiplicity $2$ but geometric multiplicity $1$. 
%If $v\in\C^n$ is an eigenvector with eigenvalue $\lambda$, then 
%$e^{\lambda t}v$  
%is a solution of $x'=Ax$. We want to find a second linearly independent solution 
%corresponding to the eigenvalue $\lambda$. 
%
%\medskip
%\begin{enumerate}
%\item
%Try another solution of the form 
%$$
%x(t)=e^{\lambda t}(ty+z), 
%$$
%with $y,z\in\C^n$. Derive  
%conditions on $y$, $z$ in order that $x$ solve the system.  
%Show that one can always find $y$, $z$ to obtain a solution of 
%this form.
%\item
%Find a fundamental set of solutions for the sytem $x'=Ax$ with
%$$
%A=\left[\begin{array}{cc} 8& 4\\ -9& -4\end{array}\right].
%$$
%\item
%Find an explicit similarity transformation to put $A$ from part (b) into
%Jordan form.
%\item
%(Not to turn in) Think about how you can generalize this: how can you
%explicitly compute the Jordan form of a matrix? How can you explicitly
%write down a fundamental set of solutions to $x'=Ax$ in general?
%\end{enumerate}

\bigskip
\item
Let
$$
A=\left[\begin{array}{ccc}
-1&0&1\\ 0&-2&0\\ 0&0&-1
\end{array}\right].
$$

\medskip
\begin{enumerate}
\item
Compute a fundamental matrix for the system $x'=Ax$.
\item
Suppose $f:\R\rightarrow\R^3$ is continuous and 
$$
\int_{-\infty}^\infty |f(t)| dt<\infty.
$$
Show that all solutions of $x'=Ax+f$ stay bounded as 
$t\rightarrow +\infty$.
\end{enumerate}

%\newpage
\item
Let $A:\R\rightarrow \R^{n\times n}$ be continuous and consider the 
matrix function $$\Phi(t)=\mbox{exp}\,\left(\int_0^t A(s) ds\right).$$ 
\begin{enumerate}
\item
Observe that $\Phi$ is a fundamental matrix for the system
$x'=A(t)x$ if either $n=1$ or $A$ is constant.
\item
Give an example for which $\Phi$ is not a fundamental matrix for 
$x'=A(t)x$.
\item
Provide, with proof, a general condition 
(for a non-constant $A$ and general $n$) that guarantees that $\Phi$ 
is a fundamental matrix.
\item
Solve the IVP 
$$
x'=A(t)x+b(t),\qquad x(0)=\left[
\begin{matrix}
1\\ 
-1
\end{matrix}\right],
$$
where 
$$b(t)=\left[
\begin{matrix}
t\\ 
t
\end{matrix}\right]\quad\mbox{and}\quad 
A(t)=\left[\begin{array}{cc}t&t^3\\ 0&t\end{array}\right]\ .$$
\end{enumerate}

\bigskip
\item
Let $A:\R\rightarrow\C^{n\times n}$ be continuous and suppose that 
$$
\liminf_{t\rightarrow\infty}\mbox{Re}\, 
\int_0^t\mbox{tr}\,(A(s))ds >-\infty\ . 
$$
Suppose that $\Phi(t)$ is a fundamental matrix for the system
$x'=A(t)x$, and suppose that $\Phi$ is uniformly bounded on $[0,\, \infty)$ 
(in some norm).
\begin{enumerate}
\item
Show that $\Phi^{-1}$ is uniformly bounded on $[0,\infty)$.
\item
Show that no non-trivial solution to $x'=A(t)x$ can satisfy
$x(t)\rightarrow 0$ as $t\rightarrow \infty$.
\end{enumerate}
\bigskip

\item 
Let $A\in\C^{n\times n}$.
Use the Spectral Mapping Theorem to prove that $\det (e^{A})=e^{\mathrm{tr} (A)}$.
%\begin{enumerate}
%\item
%Prove that $\det (e^{A})=e^{\mathrm{tr} (A)}$ by considering the  
%Wronskian of the normalized fundamental matrix for the system $x'=Ax$.
%\item
%Give a second proof using the spectral mapping theorem.
\end{enumerate}
\end{document}
\newpage
\item
A first-order autonomous system of differential equations for 
unknowns $$(x_1,\dots,x_n,p_1,\dots,p_n)=(x,p)\in\R^{2n}$$   
is said to be a {\it Hamiltonian system} if there is a real-valued function 
$H(x,p)$ on $\R^{2n}$ (usually assumed to be $C^2$),  
called the {\it Hamiltonian function}, so that the system is of the form  
$$
x_i'=\frac{\partial H}{\partial p_i}(x,p)\ ,\qquad
p_i'=-\frac{\partial H}{\partial x_i}(x,p)\ ,\qquad i=1,\dots,n\ .
$$
In mechanics, $(x_i,\, p_i)$ represents the position and momentum
of the $i{}^{th}$ particle, and $H(x,p)$ is the total energy of the
physical system.

\medskip
\begin{enumerate}
\item
Suppose $H$ is of the form
$$
H(x,p)=\frac12 \sum_{i=1}^n 
\frac{1}{m_i}p_i^2+V(x_1,\dots, x_n)\ ,
$$
where the first term represents the kinetic energy and the
second term the potential energy, assumed to depend only on the
position of the particles, and $m_i>0$ is the mass of the $i^{th}$
particle. 
Show that the system above is equivalent to Newton's equations
$$
F_i=m_ia_i
$$
for the $n$ particles, where $F_i=-\frac{\partial V}{\partial x_i}$ is the
force on the $i^{th}$ particle and $a_i=x_i''$ is its acceleration.
\item
Suppose $(x(t),p(t))$ is a solution of a Hamiltonian system.
Show that $H(x(t),p(t))$ is independent of $t$.  Thus energy is conserved:
the total energy is constant along any solution.   
\item
Suppose $f:\R^m\rightarrow\R^m$ is a $C^1$ vector field. Let $x(t,y)$ 
denote the solution of the IVP $x'=f(x),\ x(0)=y$ for $y\in\R^m$. 
The vector field $f$ is said to define a {\it volume-conserving flow}
if for each open set $\mathcal{U}\subset \R^m$ and each $t\in \R$, 
$$
\mbox{vol}\, \left(\{x(t,y):y\in \cU\}\right)=\mbox{vol}\,(\cU)\ . 
$$
It can be shown (don't do it here) that $f$ defines a volume-conserving
flow if and only if 
$$
\mbox{div}\, f\equiv \sum_{i=1}^m\frac{\partial f_i}{\partial x_i}=0\ .
$$

\noindent
Prove Liouville's Theorem: The flow defined by any Hamiltonian system
is volume conserving in $\R^{2n}$. (You may assume for simplicity that
all solutions exist for all time.)
\end{enumerate}
\end{enumerate}
\end{document}
