% Created 2016-03-21 Mon 22:35

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\usepackage{tikz}

\usepackage{minted}

\usemintedstyle{pastie}

\usetheme{default}

\author{Tobi Lehman}

\date{2016-03-02 Wed}

\title{Object Oriented Orbits: a primer on Newtonian physics}

\hypersetup{

pdfkeywords={},

pdfsubject={A tour through the mathematics, physics and source code required to simulate a universe with gravity.},

pdfcreator={Emacs 24.5.1 (Org mode 8.2.10)}}

\begin{document}

\maketitle

\begin{frame}[label=sec-1]{What does it take to simulate orbits?}

\begin{center}

\begin{tikzpicture}

\coordinate (s) at (0,0);

\coordinate (e) at (2,0);

\coordinate (m) at (3,1);

\fill[yellow] (s) circle (17pt);

\draw[black] (s) circle (17pt);

\draw[black] (e) circle (2pt);

\draw[black] (s) circle (57pt);

\node[below right] at (e) {Earth};

\fill[blue] (e) circle (2pt);

\draw[black] (m) circle (2pt);

\draw[black] (s) circle (90pt);

\node[above right] at (m) {Mars};

\fill[red] (m) circle (2pt);

\end{tikzpicture}

\end{center}

\end{frame}

\begin{frame}[label=sec-2]{What we need}

\begin{block}{Before we can simulate orbits, we need to a few things}

\begin{itemize}

\item a \alert{model of space} to organize the simulated bodies

\item a \alert{dynamic rule} to update the locations of bodies in space

\end{itemize}

\end{block}

\end{frame}

\begin{frame}[label=sec-3]{Euclid's axioms}

\begin{block}{The first complete model of space ever recorded was compiled by Euclid in ancient Greece.}

\end{block}

\end{frame}

\begin{frame}[label=sec-4]{Axiom 1}

\begin{block}{Between any two points ${\color{red} A}$ and ${\color{blue} B}$, a line segment $L$ can be drawn}

\begin{center}

\begin{tikzpicture}

\coordinate (a) at (0,0);

\coordinate (b) at (5,1);

\coordinate (L) at (2.5,0.5);

\draw (a) to (b);

\fill[red] (a) circle (2pt);

\fill[blue] (b) circle (2pt);

\node[below left] at (a) {A};

\node[below right] at (b) {B};

\node[above] at (L) {L};

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-5]{Axiom 2}

\begin{block}{A line segment $L$ can be extended indefinitely to a larger line segment $L'$, that contains $L$}

\begin{center}

\begin{tikzpicture}

\coordinate (a) at (0,0);

\coordinate (b) at (5,1);

\coordinate (L) at (2.5,0.5);

\draw (a) to (b);

\draw (-0.5,-0.1) to (5.5, 1.1);

\fill[red] (a) circle (2pt);

\fill[blue] (b) circle (2pt);

\node[below left] at (a) {A};

\node[below right] at (b) {B};

\node[above] at (L) {L};

\node[below right] at (L) {L'};

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-6]{Axiom 3}

\begin{block}{A circle can be drawn at any point with any radius}

\begin{center}

\begin{tikzpicture}

\coordinate (p) at (0,0);

\coordinate (rp) at (1.76,0);

\draw[black] (p) circle (50pt);

\fill[blue] (p) circle (2pt);

\node[below right] at (p) {P};

\draw (p) to (rp);

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-7]{Axiom 4}

\begin{block}{All right angles are congruent}

\begin{center}

\begin{tikzpicture}

\coordinate (a) at (0,0);

\coordinate (b) at (0,1);

\coordinate (c) at (1,0);

\coordinate (b') at (0,2);

\coordinate (c') at (2,0);

\draw[red, ultra thick] (a) to (b);

\draw[red, ultra thick] (a) to (c);

\draw[blue] (a) to (b');

\draw[blue] (a) to (c');

\draw (a) rectangle (0.2,0.2);

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-8]{Axiom 5 (The Parallel Postulate)}

\begin{block}{Given a line $L$ and a point $p$ not on the line, there is exactly one line $L'$ through $p$ that doesn't intersect $L$}

\begin{center}

\begin{tikzpicture}

\coordinate (L1) at (0,0);

\coordinate (L2) at (5,1);

\coordinate (L1') at (0,-1);

\coordinate (L2') at (5,0);

\coordinate (L) at (2.5, 0.5);

\coordinate (L') at (2.5, -.5);

\coordinate (p) at (4,-0.2);

\draw (L1) to (L2);

\node[above] at (L) {L};

\node[below] at (L') {L'};

\fill[black] (p) circle (2pt);

\node[below right] at (p) {p};

\draw[opacity=0.5] (L1') to (L2');

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-9]{Theorems}

\begin{block}{From these five axioms, we can deduce many useful things, the most useful for our purposes will be the Pythagorean theorem.}

\begin{center}

\begin{tikzpicture}

\coordinate (x) at (0,0);

\coordinate (y) at (0,3);

\coordinate (z) at (4,0);

\coordinate (A) at (0,1.5);

\coordinate (B) at (2,0);

\coordinate (C) at (1.5,2);

\draw[black] (x) to (y);

\draw[black] (x) to (z);

\draw[black] (y) to (z);

\draw (x) rectangle (0.5,0.5);

\node[left] at (A) {A};

\node[below] at (B) {B};

\node[right] at (C) {C};

\node[below] at (2,-1) {$A^2 + B^2 = C^2$};

\end{tikzpicture}

\end{center}

\end{block}

\begin{block}{We can use this to compute distance}

\end{block}

\end{frame}

\begin{frame}[label=sec-10]{Axioms 1 and 2 and vectors}

\begin{block}{Vectors are \alert{directed line segments}, which can be \alert{scaled by real numbers}, so axioms 1 and 2 are relevant for vectors}

\begin{enumerate}

\item Given any two points $A$ and $B$, a vector $\vec{v}$ exists whose tail is $A$ and head is $B$

\item Given any vector $\vec{v}$ and any real number $c$, $c\vec{v}$ extends $\vec{v}$ by a factor of $c$

\end{enumerate}

\begin{center}

\begin{tikzpicture}

\coordinate (a) at (0,0);

\coordinate (b) at (5,1);

\coordinate (L) at (2.5,0.5);

\draw[->,line width=2,opacity=0.7] (a) to (b);

\fill[red] (a) circle (2pt);

\fill[blue] (b) circle (2pt);

\node[below left] at (a) {A};

\node[below right] at (b) {B};

\node[above] at (L) {L};

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-11]{Some terminology}

\begin{block}{We call the initial point of a vector its \alert{tail}}

\end{block}

\begin{block}{The final point of the vector is called its \alert{head}}

\begin{center}

\begin{tikzpicture}

\coordinate (a) at (0,0);

\coordinate (b) at (2,1);

\draw[->,line width=1] (a) to (b);

\node[left,red] at (a) {tail};

\node[right,red] at (b) {head};

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-12]{Vectors can be added}

\begin{block}{Given any two vectors $\vec{v}$ and $\vec{w}$ \uline{with the same tail}, their sum $\vec{v} + \vec{w}$ can be visualized using a parallelogram:}

\begin{center}

\begin{tikzpicture}

\coordinate (a) at (0,0);

\coordinate (b) at (3,2);

\coordinate (c) at (1,3);

\coordinate (b_c) at (4,5);

\draw[->,line width=2] (a) to (b);

\draw[->,line width=2] (a) to (c);

\draw[->,line width=1] (a) to (b_c);

\draw (b) to (b_c);

\draw (c) to (b_c);

\node[left] at (c) {$\vec{v}$};

\node[below right] at (b_c) {$\vec{v}+\vec{w}$};

\node[right] at (b) {$\vec{w}$};

\end{tikzpicture}

\end{center}

\end{block}

\begin{block}{This uses \alert{axiom 5}, and this operation is commutative}

\end{block}

\end{frame}

\begin{frame}[label=sec-13]{Vectors and coordinate systems}

\begin{block}{Given a coordinate system, we can represent vectors using pairs (2D) or triples (3D) of real numbers:}

\end{block}

\begin{block}{There is a special point, $\vec{0}$ which is just the origin.}

\begin{center}

\begin{tikzpicture}

% The axes

\draw[->] (xyz cs:x=-4) -- (xyz cs:x=4) node[above] {$x$};

\draw[->] (xyz cs:y=-4) -- (xyz cs:y=4) node[right] {$z$};

\draw[->] (xyz cs:z=-4) -- (xyz cs:z=4) node[above] {$y$};

% The thin ticks

\foreach \coo in {-4,-3,...,4}

{

\draw (\coo,-1.5pt) -- (\coo,1.5pt);

\draw (-1.5pt,\coo) -- (1.5pt,\coo);

\draw (xyz cs:y=-0.15pt,z=\coo) -- (xyz cs:y=0.15pt,z=\coo);

}

% The thick ticks

\draw[thick] (3,-3pt) -- (3,3pt) node[below=6pt] {3};

\draw[thick] (-3pt,4) -- (3pt,4) node[left=6pt] {4};

\draw[thick] (xyz cs:y=-0.3pt,z=-2) -- (xyz cs:y=0.3pt,z=-2) node[below right=4pt] {-2};

\fill[blue] (2,3.5,-2) circle (2pt);

\node[left,color=blue] at (2,3.5,-2) {(2,3.5,-2)};

\draw[->,line width=1,color=red] (0,0,0) to (2,3.5,-2);

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-14]{Vectors have a 'dot product'}

\begin{block}{Given any two vectors $\vec{v} = (v_1,v_2,v_3)$ and $\vec{w} = (w_1,w_2,w_3)$}

\end{block}

\begin{block}{their dot product $\vec{v} \cdot \vec{w} = v_1w_1 + v_2w_2 + v_3w_3$}

\end{block}

\begin{block}{\alert{Useful fact:} $\vec{v} \cdot \vec{w} = |v||w|cos(\theta)$}

\end{block}

\begin{block}{That also implies that $\sqrt{\vec{v} \cdot \vec{v}}$ is the length of the vector}

\end{block}

\end{frame}

\begin{frame}[label=sec-15]{Distance between vectors}

\begin{block}{We are using vectors to represent points in space, so we will compute the distance between the points $V$ and $W$ by computing $\sqrt{(\vec{v}-\vec{w})\cdot (\vec{v}-\vec{w})}$. \alert{This dot product magic just follows from the Pythagorean theorem}.}

\begin{center}

\begin{tikzpicture}

% The axes

\draw[->] (xyz cs:x=-3) -- (xyz cs:x=3) node[above] {$x$};

\draw[->] (xyz cs:y=-3) -- (xyz cs:y=3) node[right] {$z$};

\draw[->] (xyz cs:z=-3) -- (xyz cs:z=3) node[above] {$y$};

% The thin ticks

\foreach \coo in {-3,-2,...,3}

{

\draw (\coo,-1.5pt) -- (\coo,1.5pt);

\draw (-1.5pt,\coo) -- (1.5pt,\coo);

\draw (xyz cs:y=-0.15pt,z=\coo) -- (xyz cs:y=0.15pt,z=\coo);

}

% The thick ticks

\draw[->,line width=1,color=black] (0,0,0) to (2,1,-1);

\node[right] at (2,1,-1) {$\vec{w}$};

\draw[->,line width=1,color=black] (0,0,0) to (-2,2,1);

\node[left] at (-2,2,1) {$\vec{v}$};

\node[above] at (-0.5,2,1) {$\vec{v}-\vec{w}$};

\draw[style=dashed] (2,1,-1) to (-2,2,1);

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[fragile,label=sec-16]{Vectors in Ruby (components)}

\begin{block}{Now that we have a \alert{model of space}, we can start writing some ruby code}

\begin{itemize}

\item a Vector has components (the coordinates)

\end{itemize}

\begin{minted}[]{ruby}

class Vector

attr_reader :components

def initialize(components)

@components = components

end

end

\end{minted}

\end{block}

\end{frame}

\begin{frame}[fragile,label=sec-17]{Vectors in Ruby (algebra)}

\begin{itemize}

\item a Vector can be added to another vector

\item a Vector can be multiplied by a scalar

\end{itemize}

\begin{minted}[]{ruby}

class Vector

def +(vector)

sums = components.zip(vector.components).

map {|(vi,wi)| vi+wi }

Vector.new(sum)

end

def *(scalar)

Vector.new(components.map{|c| scalar*c })

end

end

\end{minted}

\end{frame}

\begin{frame}[fragile,label=sec-18]{Vectors in Ruby (equality and dot product)}

\begin{itemize}

\item we can compare two vectors for equality

\item we can take the dot product of two vectors and get the scalar

\end{itemize}

\begin{minted}[]{ruby}

class Vector

def ==(vector)

components == vector.components

end

def dot(vector)

pairs = components.zip(vector.components)

pairs.map {|(vi,wi)| vi*wi }.

inject(&:+)

end

end

\end{minted}

\end{frame}

\begin{frame}[label=sec-19]{Time}

\begin{block}{Now we have a decent \alert{model of space}, we can move on to the \alert{dynamic rule}, it will be a way to update the state of the bodies in space over time.}

\end{block}

\end{frame}

\begin{frame}[label=sec-20]{Relation between position, time and velocity}

\begin{block}{We can represent the path a body takes using a function $\vec{x}(t)$.}

\end{block}

\begin{block}{The velocity is then just the \alert{rate of change of position with respect to time}}

\begin{center}

{\LARGE $\vec{v}(t) = \frac{d\vec{x}}{dt}$}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-21]{Relation between velocity and acceleration}

\begin{block}{Similarly, the acceleration is the \alert{rate of change of velocity with respect to time}}

\begin{center}

{\LARGE $\vec{a}(t) = \frac{d\vec{v}}{dt}$}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-22]{Newton's 1st Law states that}

\begin{block}{Bodies travel in straight lines with constant velocity \uline{unless a force is acting on it}}

\begin{center}

{\LARGE $\vec{x}(t) = \underbrace{\vec{x}_0}_\text{initial position} + \underbrace{\vec{v}_0}_\text{initial velocity}t$ }

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-23]{Newton's 2nd Law states that}

\begin{block}{The vector sum of forces acting on a body is its acceleration times its mass}

\begin{center}

{\LARGE $\underbrace{\sum_j\vec{F_{ij}}}_\text{sum of all forces acting on the i-th body} = m_i\vec{a}_i$ }

\end{center}

\end{block}

\begin{block}{Note that forces are represented as vectors}

\end{block}

\end{frame}

\begin{frame}[label=sec-24]{Newton's Law of Universal Gravitation}

\begin{center}

\begin{tikzpicture}

{\LARGE

\coordinate (a) at (0,0);

\coordinate (ahat) at (3,1.5);

\coordinate (b) at (4,2);

\draw[->,line width=1] (b) to (a);

\draw[->,line width=2] (b) to (ahat);

\node[above right] at (0.5,.5) {$ \vec{r_{ij}} $};

\node[above right] at (3,1.8) {$ \hat{\vec{r_{ij}}} $};

\node[below] at (a) {$m_i$};

\node[below right] at (b) {$m_j$};

}

\end{tikzpicture}

\end{center}

\begin{center}

{\LARGE $\vec{F_{ij}} = \left(G \frac{m_i m_j}{ |\vec{r_{ij}}|^2 }\right)\hat{\vec{r_{ij}}}$ }

\end{center}

\end{frame}

\begin{frame}[fragile,label=sec-25]{Bodies in Ruby}

\begin{block}{the Body class should have a read-only mass}

\end{block}

\begin{block}{along with a position and a velocity}

\begin{minted}[]{ruby}

class Body

attr_reader :mass

attr_accessor :position, :velocity

def initialize(mass:, position:, velocity:)

@mass = mass

@position = Vector.new(position)

@velocity = Vector.new(velocity)

end

end

\end{minted}

\end{block}

\end{frame}

\begin{frame}[fragile,label=sec-26]{Forces on Bodies in Ruby}

\begin{block}{Bodies have a method to compute the gravitational force acting on it from another Body.}

\begin{minted}[]{ruby}

class Body

def force_from(body)

rvec = body.position - position

r = rvec.norm

rhat = rvec * (1/r)

rhat * (Newtonian.G * mass * body.mass / r**2)

end

end

\end{minted}

\end{block}

\end{frame}

\begin{frame}[label=sec-27]{the Universe}

\begin{quote}

It's very big \\

- Douglas Adams

\end{quote}

\end{frame}

\begin{frame}[fragile,label=sec-28]{the Universe in Ruby}

\begin{block}{The final class will be Universe, it organizes all the bodies}

\begin{minted}[]{ruby}

class Universe

attr_reader :dimensions, :bodies

def initialize(dimensions:, bodies:)

@dimensions = dimensions

@bodies = bodies

end

end

\end{minted}

\end{block}

\begin{block}{it also has a number of \alert{\alert{dimensions}}, we can use this to make sure the bodies are all in the same kind of space}

\end{block}

\end{frame}

\begin{frame}[fragile,label=sec-29]{the Enumerable Universe}

\begin{block}{Since force is computed pairwise, we create an iterator for pairs of distinct objects}

\begin{minted}[]{ruby}

class Universe

def each_pair_with_index

bodies.each_with_index do |body_i, i|

bodies.each_with_index do |body_j, j|

next if i == j

yield [body_i, body_j, i, j]

end

end

end

end

\end{minted}

\end{block}

\end{frame}

\begin{frame}[fragile,label=sec-30]{The main simulation loop}

\begin{minted}[]{ruby}

class Universe

def evolve(dt)

forces = bodies.map{ |_| zero_vector }

each_pair_with_index do |(body_i, body_j, i, j)|

forces[i] += body_i.force_from(body_j)

end

bodies.each_with_index do |_, i|

a = forces[i] * (1.0 / bodies[i].mass)

v = bodies[i].velocity

bodies[i].velocity += a * dt

bodies[i].position += v * dt

end

end

end

\end{minted}

\end{frame}

\begin{frame}[label=sec-31]{The server}

\begin{block}{We can serve this up to a browser using}

\begin{itemize}

\item WEBrick for HTTP

\item websocketd for piping STDOUT to a WebSocket server

\end{itemize}

\end{block}

\end{frame}

\begin{frame}[fragile,label=sec-32]{Fork off an HTTP server}

\begin{minted}[]{ruby}

rd, wt = IO.pipe

pid = fork do

rd.close

server = WEBrick::HTTPServer.new({

:Port => PORT,

:BindAddress => "localhost",

:StartCallback => Proc.new {

wt.write(1) # write "1", signal start

wt.close

}

})

trap('INT') { server.stop }

server.mount("/", WEBrick::HTTPServlet::FileHandler, './examples')

server.start

end

# ...

\end{minted}

\end{frame}

\begin{frame}[fragile,label=sec-33]{Shell out to websocketd}

\begin{block}{websocketd converts standard input and output into a fully functioning websocket server, so we can just puts out the universe state}

\begin{minted}[]{ruby}

examples = ["binary.rb", "ternary.rb", "random.rb", "figure_eight.rb"]

index = ARGV.last.to_i

# Shell out to websocketd, block until program finishes

system("bin/websocketd \

-port=8080 \

ruby #{examples[index]}")

Process.kill('INT', pid) # kill HTTP server in child process

\end{minted}

\end{block}

\end{frame}

\begin{frame}[label=sec-34]{Binary Star system}

\begin{block}{Our first application is going to be simulating a binary star system, with two equal-mass stars}

\begin{center}

\begin{tikzpicture}

\coordinate (a) at (-2,0);

\coordinate (a') at (-2,-.7);

\coordinate (b) at (2,0);

\coordinate (b') at (2,.7);

\coordinate (c) at (0,0);

\draw[lightgray] (c) circle (57pt);

\draw[->,line width=1] (a) to (a');

\draw[black] (a) circle (4pt);

\fill[yellow] (a) circle (4pt);

\node[above left] at (a) {$m$};

\node[above left] at (a') {$-\vec{v}$};

\draw[->,line width=1] (b) to (b');

\draw[black] (b) circle (4pt);

\fill[yellow] (b) circle (4pt);

\node[below right] at (b) {$m$};

\node[below right] at (b') {$\vec{v}$};

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-35]{Find initial conditions}

\begin{block}{The two bodies will be traveling in uniform circular motion, so the following relation holds:}

\begin{center}

{\LARGE ${\color{red} a} = \frac{v^2}{r}$}

\begin{tikzpicture}

\coordinate (c) at (0pt,0pt);

\coordinate (p) at (75pt,0pt);

\coordinate (p') at (75pt,50pt);

\coordinate (m) at (50pt,0pt);

\draw[lightgray] (c) circle (75pt);

\draw[black] (p) circle (5pt);

\draw[->,line width=1] (p) to (p');

\draw[blue] (c) to (p);

\node[below left] at (30pt,0) {$r$};

\fill[white] (p) circle (5pt);

\node[below right] at (p') {$\vec{v}$};

\node[above right] at (p) {$|\vec{v}| = v$};

\node[below right,red] at (p) {$|\vec{a}| = a$ (centripetal acceleration)};

\draw[->,red,line width=1] (p) to (m);

\node[below,red] at (m) {$\vec{a}$};

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-36]{Given the masses and the distance $r$, we can figure out $a$:}

\begin{center}

{\LARGE $ a = (Gm^2/4r^2)/m = Gm/4r^2 $}

\end{center}

\end{frame}

\begin{frame}[label=sec-37]{Substituting a back in to get $v$}

\begin{center}

{\LARGE $v = \sqrt{(Gm/4r^2)*r} = \sqrt{Gm/4r} $ }

\end{center}

\end{frame}

\begin{frame}[label=sec-38]{Run simulated binary star system}

\begin{block}{Pause to run simulations}

\end{block}

\end{frame}

\begin{frame}[label=sec-39]{The Three Body Problem}

\begin{block}{With only two bodies, it turns out to be possible to solve the equations of motion for \alert{all time}, exactly.}

\end{block}

\begin{block}{With three or more bodies, it is in general impossible}

\end{block}

\end{frame}

\begin{frame}[label=sec-40]{However}

\begin{block}{The three body problem has been studied since 1747, and there are some well known examples}

\end{block}

\end{frame}

\begin{frame}[label=sec-41]{The "Figure Eight" Three Body Orbit}

\begin{block}{The paper "A remarkable periodic solution of the three-body problem in the case of equal masses" by Alain Chenciner and Richard Montgomery works out an orbit that looks like this:}

\begin{center}

\begin{tikzpicture}

\def\samples{100}

\def\c{1}

\pgfmathsetmacro\cc{\c*\c}

\pgfmathsetmacro\cccc{\cc*\cc}

\def\a{1.01}

\draw

(0, 0) --

plot[

variable=\t,

domain=-45+1/\samples:45-1/\samples,

samples=\samples,

smooth,

]

(\t:{\cc*sqrt(\cc*cos(2*\t) + \cc*cos(2*\t))})

-- cycle

;

\draw

(0, 0) --

plot[

variable=\t,

domain=180-45+1/\samples:180+45-1/\samples,

samples=\samples,

smooth,

]

(\t:{\cc*sqrt(\cc*cos(2*\t) + \cc*cos(2*\t))})

-- cycle

;

\path plot coordinates { (-\c, 0) (\c, 0) };

\end{tikzpicture}

\end{center}

\end{block}

\end{frame}

\begin{frame}[label=sec-42]{The initial conditions}

\begin{block}{$\vec{x_1} = [0.97000436, -0.24308753]; \vec{x_2} = -\vec{x_1}$}

\end{block}

\begin{block}{$\vec{x_3} = \vec{0}$}

\end{block}

\begin{block}{$\vec{v_3} = [-0.93240737, -0.86473146]$}

\end{block}

\begin{block}{$-2\vec{v_1} = -2\vec{v_2} = \vec{v_3}$}

\includegraphics[width=.9\linewidth]{figure8.png}

\end{block}

\end{frame}

\begin{frame}[label=sec-43]{Run simulated three-body orbit}

\begin{block}{Pause to run simulations}

\end{block}

\end{frame}

% Emacs 24.5.1 (Org mode 8.2.10)

\end{document}