Editing ThreeDimensionalConfigurations/ChallengesPt5 (section)

===New Thoughts===

In our ''Old Way of Thinking'', the hypothesized velocity flow-field was symmetric (in both directions) about the center of the elliptical trajectory.  This hypothesized Lagrangian motion isn't (and cannot be) correct because an examination of EFE's derived Riemann (Eulerian) flow-field is not symmetric about the x'-axis.  Instead, the Eulerian flow-field displays a noticeable m = 1 contribution.  Here we present an alternate hypothesis with two new features:  (1) The flow is described by circulation about an center that is shifted along the y'-axis away from the center of the ellipse; (2) The trajectory of Lagrangian fluid elements is described by motion in a cylindrical-coordinate system such that motion in the angular coordinate is uniform.

We will still insist that the trajectory of Lagrangian fluid elements is that of an ellipse described by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{x'}{x_\mathrm{surf}} \biggr]^2 + \biggl[ \frac{(y' - y'_\mathrm{center} ) }{y'_\mathrm{surf}} \biggr]^2 \, .
</math>
  </td>
</tr>
</table>
Now we will introduce a <math>~\varpi - \varphi </math> cylindrical coordinate system that is related to the x'-y' coordinate system such that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi \cos\varphi</math>
  </td>
<td align="centner">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y' - y_\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi \sin\varphi \, ,</math>
  </td>
</tr>
</table>
with <math>~|y_\varpi| < |v'_\mathrm{center}|</math>.  Mapping the other direction gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x')^2 + (y' - y_\varpi)^2</math>
  </td>
<td align="centner">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\tan\varphi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(y' - y_\varpi)}{x'} \, .</math>
  </td>
</tr>
</table>

Using the (constraint) ellipse expression to eliminate y' from these last two expressions, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(y' - y'_\mathrm{center})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y'_\mathrm{surf} \biggl[ 1 - \frac{(x')^2}{x_\mathrm{surf}^2} \biggr]^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (y' - y_\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(y'_\mathrm{center} - y_\varpi) + y'_\mathrm{surf} \biggl[ 1 - \frac{(x')^2}{x_\mathrm{surf}^2} \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x')^2 
+ 
\biggl\{ (y'_\mathrm{center} - y_\varpi) + y'_\mathrm{surf} \biggl[ 1 - \frac{(x')^2}{x_\mathrm{surf}^2} \biggr]^{1 / 2}  \biggr\}^2 \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x' \tan\varphi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(y'_\mathrm{center} - y_\varpi) + y'_\mathrm{surf} \biggl[ 1 - \frac{(x')^2}{x_\mathrm{surf}^2} \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ x' \tan\varphi + (y_\varpi - y'_\mathrm{center} )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{y'_\mathrm{surf}}{x_\mathrm{surf}} \biggl[ x_\mathrm{surf}^2 - (x')^2 \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ (x' )^2\tan^2\varphi + 2x' \tan\varphi(y_\varpi - y'_\mathrm{center} )+ (y_\varpi - y'_\mathrm{center} )^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \biggl( \frac{y'_\mathrm{surf}}{x_\mathrm{surf}} \biggr)^2 \biggl[ x_\mathrm{surf}^2 - (x')^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ (x' )^2 \biggl[ \tan^2\varphi + \biggl( \frac{y'_\mathrm{surf}}{x_\mathrm{surf}} \biggr)^2 \biggr] 
+ 
x'  \biggl[ 2\tan\varphi(y_\varpi - y'_\mathrm{center} ) \biggr] 
+ 
\biggl[ (y_\varpi - y'_\mathrm{center} )^2 - \biggl( \frac{y'_\mathrm{surf}}{x_\mathrm{surf}} \biggr)^2 x_\mathrm{surf}^2\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 
</math>
  </td>
</tr>
</table>


The roots are &hellip;
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Scratch notes:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{B}{2A} \biggl\{\pm \biggl[1 - \frac{4AC}{B^2}  \biggr]^{1 / 2} -1 \biggr\}</math>
  </td>
</tr>
</table>
where,
<div align="center">
<math>~
\frac{4AC}{B^2} 
= 
\biggl[ \underbrace{ x_\mathrm{surf}^2 \tan^2\varphi + (y'_\mathrm{surf})^2 }_{A} \biggr]
\biggl[ \underbrace{x_\mathrm{surf}^2(y_\varpi - y'_\mathrm{center} )^2 
- 
(y'_\mathrm{surf})^2 x_\mathrm{surf}^2 }_{C} \biggr]
\biggl[\underbrace{ (y_\varpi - y'_\mathrm{center} )x_\mathrm{surf}^2 \tan\varphi }_{B/2} \biggr]^{-2}
</math>
</div>

</td></tr></table>


After <math>~x'</math> has been evaluated for a given value of <math>~\varphi</math>, the accompanying value of <math>~y'</math> can be obtained, in principle, from either of the expressions:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y_\varpi + x' \tan\varphi
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; or, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~y'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{center} +
y'_\mathrm{surf} \biggl[ 1 - \frac{(x')^2}{x_\mathrm{surf}^2} \biggr]^{1 / 2} \,.
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
'''1<sup>st</sup> EXAMPLE:'''<br />  <br /><math>~x_\mathrm{surf} = 0.9200; ~~~y_\mathrm{surf} = 0.8968; ~~~y'_\mathrm{center} = 0.3501; ~~~y_\varpi = -0.1</math>
  </td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\varphi</math><br /> <br /><math>~[ \Delta\varphi = 9^\circ]</math></td>
  <td align="center" rowspan="2"><math>~A</math></td>
  <td align="center" rowspan="2"><math>~B</math></td>
  <td align="center" rowspan="2"><math>~C</math></td>
  <td align="center" rowspan="2"><math>~\frac{4AC}{B^2}</math></td>
  <td align="center" colspan="3">"plus"</td>
  <td align="center" colspan="3">"minus"</td>
  <td align="center" rowspan="2">Expression used to obtain y'</td>
</tr>
<tr>
  <td align="center" colspan="1"><math>~x'</math></td>
  <td align="center" colspan="1"><math>~y'</math></td>
  <td align="center" colspan="1"><math>~\mathrm{ATAN2}[x', (y' - y_\varpi)]</math></td>
  <td align="center" colspan="1"><math>~x'</math></td>
  <td align="center" colspan="1"><math>~y'</math></td>
  <td align="center" colspan="1"><math>~\mathrm{ATAN2}[x', (y' - y_\varpi)]</math></td>
</tr>
<tr>
  <td align="center">0.15708</td>
  <td align="center">0.82552</td>
  <td align="center">-0.12068</td>
  <td align="center">-0.50929</td>
  <td align="center">-1.1548 &times; 10<sup>+2</sup></td>
  <td align="center">-0.71575</td>
  <td align="center">+0.91355</td>
  <td align="center">-0.95593 = <math>~\varphi - \pi</math></td>
  <td align="center">+0.86194</td>
  <td align="center">+0.66367 </td>
  <td align="center">0.65616</td>
  <td align="left">
<math>~y' = y_\mathrm{center} +
y'_\mathrm{surf} \biggl[ 1 - \frac{(x')^2}{x_\mathrm{surf}^2} \biggr]^{1 / 2} \,.
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="5">&nbsp;</td>
  <td align="center" bgcolor="lightblue"><font color="white">-0.71575</font></td>
  <td align="center" bgcolor="lightblue"><font color="white">-0.21336</font></td>
  <td align="center">-2.9845 = &#x03D5; - &pi;</td>
  <td align="center" bgcolor="purple"><font color="white">+0.86194</font></td>
  <td align="center" bgcolor="purple"><font color="white">+0.03652</font></td>
  <td align="center">0.15708 = &#x03D5;</td>
  <td align="left">
<math>~y' = y_\mathrm{center} -
y'_\mathrm{surf} \biggl[ 1 - \frac{(x')^2}{x_\mathrm{surf}^2} \biggr]^{1 / 2} \,.
</math>
  </td>-0.41267</tr>
</table>




We will assume that <math>~\varphi = \dot{\varphi} t</math>, with <math>~\dot\varphi</math> constant, and then determine how <math>~\varpi</math> depends on <math>~\varphi</math> and therefore, also, how it varies with time.

First, we note that transforming from the primed-Cartesian system to the cylindrical-coordinate system is accomplished via the relations,