Editing PGE/RotatingFrame (section)

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

<span id="Utou">Alternatively,</span> setting <math>\vec{F} = \vec{X}</math> gives,
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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]\vec{X}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\vec{u} - \mathbf{\Omega^*}\vec{x} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eqs. (12) &amp; (15)</td>
</tr>
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where, adopting Chandrasekhar's notation, the variable,

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  <td align="right">
<math>\vec{u}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{d\vec{x}}{dt} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (14b)</td>
</tr>
</table>
denotes the fluid velocity as measured <font color="darkgreen">"&hellip; with respect to an observer [that is] at rest in the moving frame."</font> This allows us to write the,
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<tr>
  <td align="center" colspan="3"><font color="#770000">'''Lagrangian Representation'''</font><br />of the rotating-frame Euler Equation<br />in terms of the rotating-frame velocity, <math>\vec{u}</math></td>
</tr>

<tr>
  <td align="right">
<math>
\frac{d}{dt}\biggl[ \vec{u} - \mathbf{\Omega^*}\vec{x} \biggr] 
- \mathbf{\Omega^*}\biggl[ \vec{u} - \mathbf{\Omega^*}\vec{x} \biggr]</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>
</table>

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
Again appreciating [[#Product|from above]] that <math>(\mathbf{\Omega^*}\vec{Q})_i = \epsilon_{ijk}\Omega_k Q_j = -\epsilon_{ijk}\Omega_j Q_k</math>, in component form this version of the Euler equation reads,

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  <td align="right">
<math>\frac{d}{dt}\biggl[ u_i + \epsilon_{ijk}\Omega_j x_k  \biggr] </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\epsilon_{imk}\Omega_k \biggl[ u_m + \epsilon_{mjk}\Omega_j x_k  \biggr]
-\frac{1}{\rho} \frac{\partial p}{\partial x_i} - \frac{\partial\Phi}{\partial x_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d u_i}{dt} + \epsilon_{ijk}\biggl[\biggl( \frac{d\Omega_j}{dt} \biggr) x_k + \Omega_j \biggl( \frac{dx_k}{dt} \biggr)  \biggr] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\epsilon_{imk}u_m\Omega_k  
+
\epsilon_{imk}\Omega_k \biggl[ \epsilon_{mjk}\Omega_j x_k  \biggr]
-\frac{1}{\rho} \frac{\partial p}{\partial x_i} - \frac{\partial\Phi}{\partial x_i} 
\, .
</math>
  </td>
</tr>
</table>
Now, if we &hellip;
<ol type="a">
  <li>Swap the "jk" indices of the various terms on the LHS, which dictates that the leading sign be swapped as well:

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<tr>
  <td align="right">
<math>
\epsilon_{ijk}\biggl[\biggl( \frac{d\Omega_j}{dt} \biggr) x_k + \Omega_j \biggl( \frac{dx_k}{dt} \biggr)  \biggr] 
</math>
  </td>
  <td align="center">
<math>~~\rightarrow ~~</math>
  </td>
  <td align="left">
<math>
-~ \epsilon_{ijk}\biggl[x_j \biggl( \frac{d\Omega_k}{dt} \biggr) + u_j \Omega_k  \biggr] \, ;
</math>
  </td>
</tr>
</table>
note also that we have set <math>dx_j/dt \rightarrow  u_j</math>;
  </li>
  <li>In the first term on the RHS, replace the index, "m", with the index, "j":

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<tr>
  <td align="right">
<math>
\epsilon_{imk} u_m \Omega_k  
</math>
  </td>
  <td align="center">
<math>~~\rightarrow ~~</math>
  </td>
  <td align="left">
<math>
\epsilon_{ijk} u_j \Omega_k \, ; 
</math>
  </td>
</tr>
</table>
  </li>
  <li>Inside the square brackets of the second term on the RHS, replace the "jk" indices with "h&ell;" in order to avoid confusion, then swap the "h&ell;" indices of the two variables, which dictates that the leading sign be swapped as well:

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  <td align="right">
<math>
\biggl[ \epsilon_{mjk}\Omega_j x_k  \biggr] 
</math>
  </td>
  <td align="center">
<math>~~\rightarrow ~~</math>
  </td>
  <td align="left">
<math>
\biggl[ \epsilon_{mh\ell}\Omega_h x_\ell  \biggr] 
</math>
  </td>
  <td align="center">
<math>~~\rightarrow ~~</math>
  </td>
  <td align="left">
<math>
\biggl[-~ \epsilon_{mh\ell} x_h \Omega_\ell \biggr] \, ;
</math>
  </td>
</tr>
</table>
  </li>
  <li>Swap the "mk" indices on the Levi-Civiti tensor that lies just outside the square brackets of the second term on the RHS, which dictates that the leading sign be swapped as well:

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\epsilon_{imk}\Omega_k\biggl[-~ \epsilon_{mh\ell} x_h\Omega_\ell  \biggr] 
</math>
  </td>
  <td align="center">
<math>~~\rightarrow ~~</math>
  </td>
  <td align="left">
<math>
- ~ \epsilon_{ikm}\Omega_k\biggl[-~ \epsilon_{mh\ell} x_h \Omega_\ell \biggr] 
</math>
  </td>
  <td align="center">
<math>~~\rightarrow ~~</math>
  </td>
  <td align="left">
<math>
\epsilon_{ikm}\Omega_k\biggl[\epsilon_{mh\ell} x_h \Omega_\ell \biggr] \, ;
</math>
  </td>
</tr>
</table>
  </li>
</ol>

the Euler equation becomes,
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  <td align="right">
<math>\frac{d u_i}{dt} -~ \epsilon_{ijk}\biggl[x_j \biggl( \frac{d\Omega_k}{dt} \biggr) + u_j \Omega_k  \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\epsilon_{ijk} u_j \Omega_k  
+
\epsilon_{ikm}\Omega_k\biggl[\epsilon_{mh\ell} x_h \Omega_\ell \biggr]
-\frac{1}{\rho} \frac{\partial p}{\partial x_i} - \frac{\partial\Phi}{\partial x_i} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~\frac{d u_i}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\underbrace{2\epsilon_{ijk} u_j \Omega_k}_{[2\vec{u}\times \vec\Omega]_i}  
+
\underbrace{\epsilon_{ikm}\Omega_k\biggl[\epsilon_{mh\ell} x_h \Omega_\ell \biggr]}_{[\vec\Omega \times(\vec{x}\times\vec\Omega)]_i}
+
\underbrace{\epsilon_{ijk}\biggl[x_j \biggl( \frac{d\Omega_k}{dt} \biggr) \biggr]}_{[\vec{x} \times (d\vec\Omega/dt)]_i}
-\frac{1}{\rho} \frac{\partial p}{\partial x_i} - \frac{\partial\Phi}{\partial x_i} 
\, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

We therefore can rewrite in a more familiar ''vector'' formulation, 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 rotating-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>
2\vec{u}\times \vec\Omega + \vec\Omega \times (\vec{x}\times \vec\Omega)
+ \vec{x}\times \frac{d\vec\Omega}{dt}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{rotating} \, . 
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">{{ Rossner67 }}, &sect;II, Eq. (1)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], Appendix 1.D, &sect;3, (p. 664) Eq. (1D-42)</td>
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The RHS of this equation matches the RHS of Eq. (1) from {{ Rossner67full }} after making the notation switch, <math>\Phi \rightarrow -~\mathfrak{B}</math>, and after acknowledging that <math>\nabla P/\rho \rightarrow \nabla(P/\rho)</math> when the mass-density is spatially uniform.  The referenced equation from [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] contains all of the terms shown here, except there, the effects of pressure are ignored.

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