Editing PGE/RotatingFrame (section)

====Rotating-Frame Euler Equation====

=====Foundation=====
Suppose the 3-component vector, <math>\vec\Omega</math>, represents a general time-dependent rotation of the <math>(x_1, x_2, x_3)</math>-frame with respect to the inertial frame.  In this context, Chandrasekhar introduces a (3 &times; 3) matrix, <math>\mathbf{\Omega^*}</math>, whose nine components can be expressed in terms of the three components of <math>\vec\Omega</math> via the relations,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>(\Omega^*)_{ij}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\epsilon_{ijk}\Omega_k \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (6a)</td>
</tr>
</table>
Alternatively, we may write,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>(\Omega^*)_{ik}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\epsilon_{ikj}\Omega_j = -~\epsilon_{ijk}\Omega_j\, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="60%" 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, the six nonzero components of the matrix, <math>\mathbf{\Omega^*}</math>, are,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>(\Omega^*)_{12}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_3\, ;</math>
  </td>

<td align="center">&nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>(\Omega^*)_{13}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\Omega_2\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(\Omega^*)_{21}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\Omega_3\, ;</math>
  </td>

<td align="center">&nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>(\Omega^*)_{23}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_1\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(\Omega^*)_{31}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_2\, ;</math>
  </td>

<td align="center">&nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>(\Omega^*)_{32}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\Omega_1\, .</math>
  </td>
</tr>
</table>

----

<table border="0" align="center" cellpadding="10" width="30px">

<tr>
  <td align="center" colspan="3" bgcolor="white"><math>\mathbf{\Omega^*}</math><br /><font size="-1">(3 &times; 3 matrix)</font></td>
</tr>

<tr>
  <td align="center" width="10px" bgcolor="lightblue"><math>0</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>\Omega_3</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>-~\Omega_2</math></td>
</tr>

<tr>
  <td align="center" width="10px" bgcolor="lightblue"><math>-~\Omega_3</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>0</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>\Omega_1</math></td>
</tr>

<tr>
  <td align="center" width="10px" bgcolor="lightblue"><math>\Omega_2</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>-~\Omega_1</math></td>
  <td align="center" width="10px" bgcolor="lightblue"><math>0</math></td>
</tr>
</table>

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

<span id="Product">For later use,</span> we note as well that for an arbitrary vector &#8212; call it, <math>\vec{Q}</math> &#8212; the individual components of the product, <math>\mathbf{\Omega^*} \vec{Q}</math>, are given by the expression,

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

<tr>
  <td align="right">
<math>(\mathbf{\Omega^*}\vec{Q} )_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
(\Omega^*)_{ij}Q_j
=
(\epsilon_{ijk}\Omega_k)Q_j
\, .
</math>
  </td>
</tr>
</table>
Compare, for example, Eqs. (17) and (19) in &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].

Now, if the motion of the moving frame relative to the inertial frame is specified entirely by the vector <math>\vec\Omega</math>, Chandrasekhar proves that any time-dependent vector defined in the inertial frame &#8212; call it <math>\vec{F}</math> &#8212; will obey the following operator relation:
<table border="0" align="center" cellpadding="5">

<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{F}
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (11)</td>
</tr>
</table>

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

For example, if we set <math>\vec{F} = d\vec{X}/dt</math>, we find,
<table border="0" align="center" cellpadding="5">

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

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

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

<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>

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

<span id="Utou">Alternatively,</span> setting <math>\vec{F} = \vec{X}</math> gives,
<table border="0" align="center" cellpadding="5">

<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>
</table>

where, adopting Chandrasekhar's notation, the variable,

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

<tr>
  <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,
<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}{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,

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

<tr>
  <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:

<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>
</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>
</tr>
</table>
----

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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