Editing PGE/RotatingFrame (section)

=====Lagrangian Representation In Terms of Inertial-Frame Velocities=====

For example, if we set <math>\vec{F} = d\vec{X}/dt</math>, we find,
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<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[
\mathbf{T}\frac{d}{dt} 
-
\biggl(\frac{d}{dt} - \mathbf{\Omega^*}\biggr)\mathbf{T}
\biggr] \frac{d\vec{X}}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \mathbf{T} \frac{d^2\vec{X}}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl(\frac{d}{dt} - \mathbf{\Omega^*}\biggr)\mathbf{T}
\frac{d\vec{X}}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{d\vec{U}}{dt} - \mathbf{\Omega^*}\vec{U} \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eqs. (13) &amp; (16)</td>
</tr>
</table>
This allows us to write the,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Lagrangian Representation'''</font><br />of the rotating-frame Euler Equation<br />in terms of the (transformed) inertial-frame velocity, <math>\vec{U}</math></td>
</tr>
<tr>
  <td align="right">
<math>\frac{d\vec{U}}{dt} - \mathbf{\Omega^*}\vec{U}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{rotating} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d\vec{U}}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\mathbf{\Omega^*}\vec{U}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{rotating} 
\, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (17)</td>
</tr>
</table>


<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
Appreciating [[#Product|from above]] that <math>(\mathbf{\Omega^*}\vec{Q})_i = \epsilon_{ijk}\Omega_k Q_j</math>, in component form this version of the Euler equation reads,
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<tr>
  <td align="right">
<math>\frac{dU_i}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\epsilon_{imk}\Omega_k U_m -\frac{1}{\rho} \frac{\partial p}{\partial x_i} - \frac{\partial\Phi}{\partial x_i} \, .</math>
  </td>
</tr>
</table>
That is,
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<tr>
  <td align="right">Component #1:&nbsp; &nbsp;</td>
  <td align="right">
<math>\frac{dU_1}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_3 U_2 - \Omega_2 U_3
-\frac{1}{\rho} \frac{\partial p}{\partial x_1} - \frac{\partial\Phi}{\partial x_1} </math>
  </td>
</tr>

<tr>
  <td align="right">Component #2:&nbsp; &nbsp;</td>
  <td align="right">
<math>\frac{dU_2}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_1 U_3 - \Omega_3 U_1
-\frac{1}{\rho} \frac{\partial p}{\partial x_2} - \frac{\partial\Phi}{\partial x_2} </math>
  </td>
</tr>

<tr>
  <td align="right">Component #3:&nbsp; &nbsp;</td>
  <td align="right">
<math>\frac{dU_3}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_2 U_1 - \Omega_1 U_2 
-\frac{1}{\rho} \frac{\partial p}{\partial x_3} - \frac{\partial\Phi}{\partial x_3} </math>
  </td>
</tr>
</table>

Notice as well that the individual components of the cross product of <math>\vec{U}</math> and <math>\vec\Omega</math> can be represented by the same summation expression, that is,
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<tr>
  <td align="right">
<math>(\vec{U} \times \vec\Omega)_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\epsilon_{imk}\Omega_k U_m \, .</math>
  </td>
</tr>
</table>

This allows us to rewrite the,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Lagrangian Representation'''</font><br />of the rotating-frame Euler Equation<br />in terms of the (transformed) inertial-frame velocity, <math>\vec{U}</math></td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\vec{U}}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\vec\Omega \times \vec{U}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{rotating} 
\, .
</math>
  </td>
</tr>
</table>
in what is perhaps more recognizable notation.

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