Editing ThreeDimensionalConfigurations/RiemannStype (section)

==Preferred Normalizations==

Let's do this again, assuming that <math>~x</math> and <math>~y</math> both have units of length and that <math>~t</math> has the unit of time.  Then, let's use <math>~a</math> to normalize lengths and use <math>~(\pi G \rho)^{-1 / 2}</math> to normalize time. We therefore have,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{x}{a} = \biggl(\frac{ x_\mathrm{max} }{a}\biggr) \cos\biggl[ \frac{\varphi}{(\pi G \rho)^{1 / 2}}  \cdot \frac{t}{(\pi G \rho)^{-1 / 2}} \biggr]
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
\frac{y}{a} = \biggl(\frac{ y_\mathrm{max} }{a}\biggr) \sin\biggl[ \frac{\varphi}{(\pi G \rho)^{1 / 2}}  \cdot \frac{t}{(\pi G \rho)^{-1 / 2}} \biggr] \, .
</math>
  </td>
</tr>
</table>


<table border="1" cellpadding="10" width="80%" align="center"><tr><td align="left">
<font color="red">'''NOTE:'''</font> &nbsp; When implementing in an xml-based COLLADA (3D animation) file, we associate <math>~\mathrm{TIME} = 4</math> with <math>~t \cdot (\pi G \rho)^{1 / 2} = 2\pi</math>.  Hence we can everywhere replace <math>~t \cdot (\pi G \rho)^{1 / 2}</math> with (in ''radians'') <math>~(\pi/2)\cdot \mathrm{TIME}</math> or (in ''degrees'') <math>~90 \cdot \mathrm{TIME}</math>.

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This also means that, if <math>~\varphi/(\pi G \rho)^{1 / 2} = 1</math>,  each Lagrangian fluid element will move through one complete orbit (as viewed from a frame that is rotating with the ellipsoidal figure) in the time it takes the hand of the wall-mounted clock to complete one cycle.
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Next, let's normalize the velocities such that <math>~\rho</math> and the total mass, <math>~M</math>, are both assumed to be the same in every examined Riemann ellipsoid.  In particular, we will normalize to,
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  <td align="right">
<math>~v_0</math>
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  <td align="center">
<math>~\equiv</math>
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  <td align="left">
<math>~(abc)^{1 / 3}(\pi G \rho)^{1 / 2}</math>
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  <td align="center">
<math>~=</math>
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<math>~a(\pi G \rho)^{1 / 2} \biggl( \frac{b}{a}\cdot \frac{c}{a} \biggr)^{1 / 3} \, ,</math>
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</table>
in which case we have,

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<tr>
  <td align="right">
<math>
\frac{1}{v_0} \cdot \frac{dx}{dt} =  - \frac{\varphi}{(\pi G \rho)^{1 / 2} } \biggl(\frac{a}{b}\biggr)\biggl( \frac{b}{a}\cdot \frac{c}{a} \biggr)^{-1 / 3} \cdot \biggl(\frac{y}{a}\biggr)
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
\frac{1}{v_0} \cdot \frac{dy}{dt} =  + \frac{\varphi}{(\pi G \rho)^{1 / 2} }  \biggl(\frac{b}{a}\biggr) \biggl( \frac{b}{a}\cdot \frac{c}{a} \biggr)^{-1 / 3}  \cdot \biggl(\frac{x}{a}\biggr) \, .
</math>
  </td>
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</table>
Finally, setting, <math>~\varphi/(\pi G\rho)^{1 / 2} \rightarrow -\lambda_\mathrm{EFE}</math> means,

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  <td align="right">
<math>
V_x \equiv \frac{1}{v_0} \cdot \frac{dx}{dt} =  \lambda_\mathrm{EFE} \biggl(\frac{a}{b}\biggr)\biggl( \frac{b}{a}\cdot \frac{c}{a} \biggr)^{-1 / 3} \cdot \biggl(\frac{y}{a}\biggr)
</math>
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  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
V_y \equiv \frac{1}{v_0} \cdot \frac{dy}{dt} = - \lambda_\mathrm{EFE}  \biggl(\frac{b}{a}\biggr) \biggl( \frac{b}{a}\cdot \frac{c}{a} \biggr)^{-1 / 3}  \cdot \biggl(\frac{x}{a}\biggr) \, .
</math>
  </td>
</tr>
</table>