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<title>Object Oriented Orbits: a primer on Newtonian physics</title>
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<div id="content">
<h1 class="title">Object Oriented Orbits: a primer on Newtonian physics</h1>
<div id="outline-container-sec-1" class="outline-2">
<h2 id="sec-1"><span class="section-number-2">1</span> We need to go to Mars</h2>
<div class="outline-text-2" id="text-1">
<p>
(Arnold Schwartzeneggar reference)
</p>
<p>
Before we can do this, we need to simulate the orbits of Earth and Mars.
</p>
</div>
</div>
<div id="outline-container-sec-2" class="outline-2">
<h2 id="sec-2"><span class="section-number-2">2</span> What we need</h2>
<div class="outline-text-2" id="text-2">
</div><ol class="org-ol"><li><a id="sec-2-1" name="sec-2-1"></a>Before we can simulate orbits, we need to a few things<br /><div class="outline-text-3" id="text-2-1">
<ul class="org-ul">
<li>a <b>model of space</b>
</li>
<li>a <b>dynamic rule</b> to update the locations of bodies in space
</li>
</ul>
</div>
</li></ol>
</div>
<div id="outline-container-sec-3" class="outline-2">
<h2 id="sec-3"><span class="section-number-2">3</span> Euclid's axioms</h2>
<div class="outline-text-2" id="text-3">
</div><ol class="org-ol"><li><a id="sec-3-1" name="sec-3-1"></a>The first complete model of space ever recorded was compiled by Euclid in ancient Greece.<br /></li></ol>
</div>
<div id="outline-container-sec-4" class="outline-2">
<h2 id="sec-4"><span class="section-number-2">4</span> Axiom 1</h2>
<div class="outline-text-2" id="text-4">
</div><ol class="org-ol"><li><a id="sec-4-1" name="sec-4-1"></a>Between any two points \({\color{red} A}\) and \({\color{blue} B}\), a line segment \(L\) can be drawn<br /><div class="outline-text-3" id="text-4-1">
</div>
</li></ol>
</div>
<div id="outline-container-sec-5" class="outline-2">
<h2 id="sec-5"><span class="section-number-2">5</span> Axiom 2</h2>
<div class="outline-text-2" id="text-5">
</div><ol class="org-ol"><li><a id="sec-5-1" name="sec-5-1"></a>A line segment \(L\) can be extended indefinitely to a larger line segment \(L'\), that contains \(L\)<br /><div class="outline-text-3" id="text-5-1">
</div>
</li></ol>
</div>
<div id="outline-container-sec-6" class="outline-2">
<h2 id="sec-6"><span class="section-number-2">6</span> Axiom 3</h2>
<div class="outline-text-2" id="text-6">
</div><ol class="org-ol"><li><a id="sec-6-1" name="sec-6-1"></a>A circle can be drawn at any point with any radius<br /><div class="outline-text-3" id="text-6-1">
</div>
</li></ol>
</div>
<div id="outline-container-sec-7" class="outline-2">
<h2 id="sec-7"><span class="section-number-2">7</span> Axiom 4</h2>
<div class="outline-text-2" id="text-7">
</div><ol class="org-ol"><li><a id="sec-7-1" name="sec-7-1"></a>All right angles are congruent<br /><div class="outline-text-3" id="text-7-1">
</div>
</li></ol>
</div>
<div id="outline-container-sec-8" class="outline-2">
<h2 id="sec-8"><span class="section-number-2">8</span> Axiom 5</h2>
<div class="outline-text-2" id="text-8">
</div><ol class="org-ol"><li><a id="sec-8-1" name="sec-8-1"></a>Given a line \(L\) and a point \(p\) not on the line, there is exactly one line \(L'\) through \(p\) that doesn't intersect \(L\)<br /><div class="outline-text-3" id="text-8-1">
</div>
</li></ol>
</div>
<div id="outline-container-sec-9" class="outline-2">
<h2 id="sec-9"><span class="section-number-2">9</span> Theorems</h2>
<div class="outline-text-2" id="text-9">
<p>
From these five axioms, we can deduce many useful things, the most useful for our
purposes will be the Pythagorean theorem.
</p>
</div>
</div>
<div id="outline-container-sec-10" class="outline-2">
<h2 id="sec-10"><span class="section-number-2">10</span> An algebraic approach to space</h2>
<div class="outline-text-2" id="text-10">
</div><ol class="org-ol"><li><a id="sec-10-1" name="sec-10-1"></a>The ancient Greek tools for doing calculations with space were a <b>compass</b> and <b>straightedge</b><br /><div class="outline-text-3" id="text-10-1">
<div class="figure">
<p><img src="./disapprove.png" alt="disapprove.png" />
</p>
</div>
</div>
</li>
<li><a id="sec-10-2" name="sec-10-2"></a>Since we are in the future, we can use <b>vectors</b> and <b>computers</b> instead<br /></li></ol>
</div>
<div id="outline-container-sec-11" class="outline-2">
<h2 id="sec-11"><span class="section-number-2">11</span> Axioms 1 and 2 and vectors</h2>
<div class="outline-text-2" id="text-11">
</div><ol class="org-ol"><li><a id="sec-11-1" name="sec-11-1"></a>Vectors are <b>directed line segments</b>, with an outer multiplcation by real numbers, so we can use axioms 1 and 2<br /><div class="outline-text-3" id="text-11-1">
<ol class="org-ol">
<li>Given any two points \(A\) and \(B\), a vector \(\vec{v}\) exists whose tail is \(A\) and head is \(B\)
</li>
<li>Given any vector \(\vec{v}\) and any real number \(c\), \(c\vec{v}\) extends \(\vec{v}\) by a factor of \(c\)
</li>
</ol>
</div>
</li></ol>
</div>
<div id="outline-container-sec-12" class="outline-2">
<h2 id="sec-12"><span class="section-number-2">12</span> Some terminology</h2>
<div class="outline-text-2" id="text-12">
</div><ol class="org-ol"><li><a id="sec-12-1" name="sec-12-1"></a>We call the initial point of a vector its <b>tail</b><br /></li>
<li><a id="sec-12-2" name="sec-12-2"></a>The final point of the vector is called its <b>head</b><br /><div class="outline-text-3" id="text-12-2">
</div>
</li></ol>
</div>
<div id="outline-container-sec-13" class="outline-2">
<h2 id="sec-13"><span class="section-number-2">13</span> Vectors can be added</h2>
<div class="outline-text-2" id="text-13">
</div><ol class="org-ol"><li><a id="sec-13-1" name="sec-13-1"></a>Given any two vectors \(\vec{v}\) and \(\vec{w}\) <span class="underline">with the same tail</span>, their sum \(\vec{v} + \vec{w}\) can be visualized using a parallelogram:<br /><div class="outline-text-3" id="text-13-1">
</div>
</li></ol>
</div>
<div id="outline-container-sec-14" class="outline-2">
<h2 id="sec-14"><span class="section-number-2">14</span> Vectors and coordinate systems</h2>
<div class="outline-text-2" id="text-14">
</div><ol class="org-ol"><li><a id="sec-14-1" name="sec-14-1"></a>Given a coordinate system, we can represent vectors using pairs (2D) or triples (3D) of real numbers:<br /></li></ol>
</div>
<div id="outline-container-sec-15" class="outline-2">
<h2 id="sec-15"><span class="section-number-2">15</span> Vectors, points and you</h2>
<div class="outline-text-2" id="text-15">
<p>
If you pick a point, call it \(O\), and then consider the set of all vectors \(\vec{OP}\) where \(P\) is some point,
you can see that there is a natural correspondence between points and vectors.
</p>
<p>
DIAGRAM
</p>
</div>
</div>
<div id="outline-container-sec-16" class="outline-2">
<h2 id="sec-16"><span class="section-number-2">16</span> Points in Ruby</h2>
<div class="outline-text-2" id="text-16">
</div><ol class="org-ol"><li><a id="sec-16-1" name="sec-16-1"></a>We can represent points using arrays of real numbers<br /></li>
<li><a id="sec-16-2" name="sec-16-2"></a>If \(A\) and \(B\) are points, then \(A-B\) is the vector from \(B\) to \(A\):<br /><div class="outline-text-3" id="text-16-2">
<div class="org-src-container">
<pre class="src src-ruby">A = Point.new([ax,ay,az])
B = Point.new([bx,by,bz])
(A-B).is_a?(Vector)
# => true
</pre>
</div>
</div>
</li></ol>
</div>
<div id="outline-container-sec-17" class="outline-2">
<h2 id="sec-17"><span class="section-number-2">17</span> Vectors in Ruby</h2>
<div class="outline-text-2" id="text-17">
<p>
So we will need a Ruby class that represents a vector, as these can be used to model
positions.
</p>
<div class="org-src-container">
<pre class="src src-ruby">class Vector
attr_reader :components
end
</pre>
</div>
</div>
</div>
<div id="outline-container-sec-18" class="outline-2">
<h2 id="sec-18"><span class="section-number-2">18</span> Relation between position, time and veloctiy</h2>
<div class="outline-text-2" id="text-18">
</div><ol class="org-ol"><li><a id="sec-18-1" name="sec-18-1"></a>Considering <b>Time</b>, we can represent the path a body takes using a function \(\vec{x}(t)\).<br /></li>
<li><a id="sec-18-2" name="sec-18-2"></a>The velocity is then just the <b>rate of change of position with respect to time</b><br /><div class="outline-text-3" id="text-18-2">
<p>
\(\vec{v}(t) = \frac{d\vec{x}}{dt}\)
</p>
</div>
</li></ol>
</div>
<div id="outline-container-sec-19" class="outline-2">
<h2 id="sec-19"><span class="section-number-2">19</span> Relation between velocity and acceleration</h2>
<div class="outline-text-2" id="text-19">
</div><ol class="org-ol"><li><a id="sec-19-1" name="sec-19-1"></a>Similarly, the acceleration is the <b>rate of change of velocity with respect to time</b><br /></li>
<li><a id="sec-19-2" name="sec-19-2"></a>\(\vec{a}(t) = \frac{d\vec{v}}{dt}\)<br /></li></ol>
</div>
<div id="outline-container-sec-20" class="outline-2">
<h2 id="sec-20"><span class="section-number-2">20</span> Enter Mr. Newton</h2>
<div class="outline-text-2" id="text-20">
</div><ol class="org-ol"><li><a id="sec-20-1" name="sec-20-1"></a>Newton's 1st Law states that \(\vec{F} = m\vec{a}\)<br /></li></ol>
</div>
</div>
<div id="postamble" class="status">
<p class="date">Date: 2016-03-02 Wed</p>
<p class="author">Author: Tobi Lehman</p>
<p class="date">Created: 2016-03-04 Fri 15:17</p>
<p class="creator"><a href="http://www.gnu.org/software/emacs/">Emacs</a> 24.5.1 (<a href="http://orgmode.org">Org</a> mode 8.2.10)</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>