Editing PGE/RotatingFrame (section)

===Part B===
Drawing from Chapter 4, &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; where the Cartesian components of the inertial-frame velocity <math>(\vec{v}_\mathrm{inertial})</math> are represented by <math>U_i</math> and the Cartesian components of the rotating-frame velocity <math>(\vec{v}_\mathrm{rot})</math> are represented by <math>u_i</math> &#8212; we begin by restating the [[PGE/Euler#in_terms_of_velocity|Lagrangian representation of the intertial-frame Euler equation]]:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\frac{dU_i}{dt}\biggr|_\mathrm{inertial}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{1}{\rho} \frac{\partial p}{\partial x_i} - \frac{\partial\Phi}{\partial x_i} \, .</math>
  </td>
</tr>
</table>

The LHS of this (Euler) equation transform as follows:
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\frac{dU_i}{dt}\biggr|_\mathrm{inertial}</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>\frac{dU_i}{dt}\biggr|_\mathrm{rot} - \epsilon_{imk}\Omega_k U_m\, ,</math>
  </td>
</tr>
</table>
where we also recognize that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>U_i</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>u_i + \epsilon_{ijk}\Omega_j x_k \, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">

Both of these expressions make use of the three-element [https://en.wikipedia.org/wiki/Levi-Civita_symbol#Definition Levi-Civita tensor], <math>\epsilon_{ijk}</math>. Its six nonzero component values are &hellip;
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center"><math>ij k</math></td>
  <td align="center"><math>\epsilon_{ijk}</math></td>
  <td rowspan="4" bgcolor="lightgrey">&nbsp;</td>
  <td align="center"><math>ij k</math></td>
  <td align="center"><math>\epsilon_{ijk}</math></td>
</tr>
<tr>
  <td align="center">123</td>
  <td align="center" rowspan="3">+1</td>
  <td align="center">132</td>
  <td align="center" rowspan="3">-1</td>
</tr>
<tr>
  <td align="center">312</td>
  <td align="center">321</td>
</tr>
<tr>
  <td align="center">231</td>
  <td align="center">213</td>
</tr>
</table> 

Hence, for example, transforming the x-component <math>(i=1)</math> of <math>\vec{U}</math> gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>U_1</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>u_1 + \epsilon_{1jk}\Omega_j x_k = u_1 + \epsilon_{123}\Omega_2 x_3 + \epsilon_{132}\Omega_3 x_2 
= u_1 + \Omega_2 z -\Omega_3 y \, ;</math>
  </td>
</tr>
</table>
transforming the y-component <math>(i=2)</math> gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>U_2</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>u_2 + \epsilon_{2jk}\Omega_j x_k = u_2 + \epsilon_{231}\Omega_3 x_1 + \epsilon_{213}\Omega_1 x_3 
= u_2 + \Omega_3 x -\Omega_1 z \, ;</math>
  </td>
</tr>
</table>
and transforming the z-component <math>(i=3)</math> gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>U_3</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>u_3 + \epsilon_{3jk}\Omega_j x_k = u_3 + \epsilon_{312}\Omega_1 x_2 + \epsilon_{321}\Omega_2 x_1 
= u_3 + \Omega_1 y -\Omega_2 x \, .</math>
  </td>
</tr>
</table>

These are the same three components that arise from the vector expression (from above),
<div align="center">
<math>
\vec{v}_{inertial} = \vec{v}_{rot} + {\vec{\Omega}} \times \vec{x} \, ;
</math>
</div>
we therefore recognize that, <math>\vec{\Omega} \times \vec{x} = \epsilon_{ijk}\Omega_j x_k</math>. We note as well that, <math>\vec{\Omega} \times \vec{x} = -\epsilon_{ijk}\Omega_k x_j</math>. 


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

Therefore, as viewed from the rotating frame of reference, the Euler equation becomes,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\frac{dU_i}{dt}\biggr|_\mathrm{rot} - \epsilon_{imk}\Omega_k U_m</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\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}{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:

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

<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":

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

<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:

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

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

<tr>
  <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>