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\noindent
\Large Math 555\hfill Winter 2015

\noindent
{\bf Homework Set 2\hfill Due Friday January 16}
\bigskip

\begin{enumerate}
\item
Given a function $p(t,x,\xi)\in C^2([a,b]\times \bRn\times\bRn)$, the Hamiltonian 
flow for $p$ is the system in $(x,\xi)$:
\[
x'(t)=(\nabla_\xi p)(t,x(t),\xi(t)),\quad \xi'(t)=-(\nabla_x p)(t,x(t),\xi(t)).
\]
\begin{enumerate}
\item
Use the change of variables formula to show that the corresponding 
flow maps $S_{t_0}^{t_1}:\bR^{2n}\times\bR^{2n}$
preserve measure; that is,
\[
\int f(S_{t_0}^{t_1}(x,\xi))\; dx\; d\xi\ =\ \int f(x,\xi)\; dx\; d\xi\ ,
\]
where you may take f to be continuous and supported in a compact set.
\item
Show that
\(
\partial_t(p(t,x(t),\xi(t)))=(\partial_tp)(t,x(t),\xi(t)).
\)
Conclude that if $p$ is
autonomous, then $p(x(t),\xi(t))$ is constant in t.
\item
For a body falling on a straight line towards the earth, with $x$ the distance
from the center of the earth, and $\xi$ the velocity of the object (negative is
downwards) then $(x,\xi)$ satisfies the Hamiltonian 
flow with
\[
p(x,\xi)=\half\xi^2-\frac{gM}{x},
\]
where $g$ is the gravitational constant and $M$ is the mass of the earth.
Use part (b) to derive an equation $\frac{d\xi}{dt}=f(\xi)$ for 
an object dropped from initial
position $x=R$ and initial velocity $0$.
\end{enumerate}
\item
Suppose that $A(t)$ is a continuous function from an interval $I$ into $\Cnn$ such
that $A(t)$ skew symmetric for all $t\in I$ ($A(t)=-A^*(t)$).
\begin{enumerate}
\item
If $\Phi(t)$ is a fundamental matrix for $x'=Ax$ on $I$ normalized at $t_0\in I$,
show that $\Phi(t)$ is unitary for all $t\in I$ . In particular, if $A(t)$ is real valued,
then $\Phi(t)$ is an orthogonal matrix.
\item
In general, if $Q\in\Rnn$, and $Q(v,w)$ is the quadratic form on $\Cn$ given by
\[
Q(v,w)=\sum_{j=1}^n(Qv)_j\bar w_j,
\]
show that $Q(\Phi(t)v,\Phi(t)w)=Q(v,w)$ for all vectors $v$ and $w$ provided that
\[
A^*(t)Q+QA(t)=0\quad\forall\, t\in I.
\]
\item
On an even dimensional space $\R^{2n}$, let $Q$ be the matrix in block form
\[
Q=\left[ \begin{array}{cc}0&-I_n\\ I_n& 0\end{array}\right]\ .
\]
The corresponsing skew-symmetric form on $\R^{2n}$ is the {\it symplectic form}, i.e.,
\[Q\left(\begin{array}{c}x\\ y\end{array}\right)=-\left(\begin{array}{c}y\\ x\end{array}\right),\]
and a matrix $M$ preserving this form (i.e. $M^TQM=Q$) is called a {\it symplectic matrix}.
Expressing $A(t)$ in block form
\[
A(t)=\left[ \begin{array}{cc}A_1(t)&A_2(t)\\ A_3(t)& A_4(t)\end{array}\right]\ ,
\]
where the blocks are $n\times n$ derive a condition on the $A_j(t)$ such that
the above holds; that is, so that $\Phi(t)$ is symplectic. 
Show that the linearized Hamiltonian 
flow matrix of problem 1 above is symplectic. 
That is, $\frac{D(x,\xi)}{D(y,\eta)}$ is symplectic, for each $(y,\eta)$ and $t$, where
$x(t_0)=y,\ \xi(t_0)=\eta$ and $(x,\xi)\in\R^{2n}$.
\end{enumerate}
Remark: For background on symplectic forms see
{\small\begin{verbatim}
https://web.math.princeton.edu/~templier/junior/5/Symplectic_Matrices_Talk.pdf
\end{verbatim}}
\item
Let $A\in\Cnn$ and let $\lambda\in\C$ be an eigenvalue of $A$ of algebraic multiplicity
$2$ but geometric multiplicity $1$. 
If $v\in\Cn$ 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.
\begin{enumerate}
\item
Try another solution of the form
\[
x(t):=e^{\lambda t}(ty+z)\quad\mbox{with  }y,z\in\Cn.
\]
Derive conditions on $(y,z)$ in order that this $x$ solves 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.
\end{enumerate}
\item
Let $A$ be the matrix
\[
A=\left[\begin{array}{ccc} -1&-1&1\\ 1&-1&-1\\2&0&-2\end{array}\right].
\]
\begin{enumerate}
\item
Find the Jordan decomposition of $A$.
\item
Find a fundamental matrix for the initial value problem $x'=Ax,\ x(0)=x_0$.
\item
Let $f:\R^3\to\R^3$ be a continuous vector-valued function with
$\int_0^\infty |f(t)|dt <\infty.$
Let $x(t)$ be the solution to the following initial value problem for $t\ge 0$:
\[
x'(t)=Ax(t)+f(t),\quad x(0)=0.
\]
Show there exists a constant vector $v\in\R^3$ such that
$|x(t)-v|\to 0$ as $t\to+\infty$, and calculate $v$ in terms of $f$ (Your answer is allowed to
involve the inverse of a matrix.)
\end{enumerate}
\end{enumerate}
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