Editing PGE/RotatingFrame

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=Rotating Reference Frame=
==Overview==
<span id="InertialFrame">Among</span> the [[PGE#Principal_Governing_Equations|principal governing equations]] we have included the
&nbsp;<br />
&nbsp;<br />

<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the inertial-frame Euler Equation,

{{Template:Math/EQ_Euler01}}

[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 671, Appendix Eq. (1E-6)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.55)
</div>
Alternatively, a rewrite of the LHS gives what we refer to as the,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the intertial-frame Euler Equation
  </td>
</tr>
<tr>
  <td align="right">
<math>
\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{1}{\rho}\nabla P - \nabla \Phi \, . 
</math>
  </td>
</tr>
</table>
At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with an angular velocity
<div align="center"> 
<math>\vec{\Omega} = \hat\imath ~\Omega_1 + \hat\jmath ~\Omega_2 + \hat{k}~\Omega_3 \, .</math> 
</div>

----

<table border="0" align="center" width="90%" cellpadding="8"><tr><td align="left">Often it suffices to align <math>\vec\Omega</math> with the z-axis of the chosen coordinate system &#8212; in which case, <math>\Omega_1 =\Omega_2 =0</math> &#8212; and to set <math>d\vec\Omega/dt = 0</math>, in which case the nonzero component of the frame's angular velocity, <math>\Omega_3</math>, is independent of time.</td></tr></table>

----

In what follows we show that, when viewed from this rotating reference frame, we have what will be referred to as 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</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>
where the difference between the rotating-frame velocity, <math>\vec{u}</math>, and the inertial-frame velocity, <math>\vec{v}</math>, is given by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\vec{v} - \vec{u}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\vec\Omega \times \vec{x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\hat\imath (\Omega_2 z - \Omega_3 y) 
+ \hat\jmath (\Omega_3 x - \Omega_1 z) 
+ \hat{k} (\Omega_1 y - \Omega_2 x)) 
\, .
</math>
  </td>
</tr>
</table>

As above, a rewrite of the LHS gives what we will refer to as the, 
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Eulerian Representation'''</font><br />of the ''rotating-frame'' Euler Equation</td>
</tr>

<tr>
  <td align="right">
<math>
\frac{\partial\vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u}
</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>
</table>

Along the way, and being guided by Chandrasekhar's presentation in Chapter 4, &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we appreciate that it can be useful to highlight a ''hybrid'' representation of the Euler Equation that involves a mixture of the velocity variables, <math>\vec{u}</math> along with <math>\vec{v}</math>.  For example, beginning with this last expressions, we can write,

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

<tr>
  <td align="right">
<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] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial\vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial}{\partial t} \biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]
+ 
(\vec{u} \cdot \nabla)\biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+ \frac{\partial}{\partial t} \biggl[- \vec\Omega \times \vec{x} \biggr]
+ 
(\vec{u} \cdot \nabla)\biggl[- \vec\Omega \times \vec{x} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
- \frac{d}{dt}\biggl[\vec\Omega \times \vec{x} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+
\vec{x}\times \frac{d\vec\Omega}{\partial t} - \vec\Omega \times \vec{v}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+
\vec{x}\times \frac{d\vec\Omega}{dt} - \vec\Omega \times \biggl[\vec{u} + \vec\Omega \times \vec{x}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+
\vec{x}\times \frac{d\vec\Omega}{dt} 
+ \vec{u}\times \vec\Omega
+ \vec\Omega \times (\vec{x}\times \vec\Omega )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\vec{u}\times \vec\Omega 
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr] 
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>-\frac{1}{\rho}\nabla P - \nabla \Phi</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)[\vec{u} + \vec\Omega \times \vec{x}]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} 
+ (\vec{v}\cdot \nabla)\vec{u}
+ (\vec{v}\cdot \nabla)[ \vec\Omega \times \vec{x}]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}
+ \biggl[ v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y} + v_z \frac{\partial}{\partial z} \biggr]
\biggl[ 
\hat\imath (\Omega_2 z - \Omega_3 y) 
+ \hat\jmath (\Omega_3 x - \Omega_1 z) 
+ \hat{k} (\Omega_1 y - \Omega_2 x)) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}
+ v_x
\biggl[ 
\hat\jmath (\Omega_3 ) 
- \hat{k} (\Omega_2 ) 
\biggr]
+ v_y 
\biggl[ 
-\hat\imath (\Omega_3) 
+ \hat{k} (\Omega_1 )
\biggr]
+ v_z 
\biggl[ 
\hat\imath (\Omega_2 ) 
- \hat\jmath (\Omega_1) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}
+ \underbrace{\hat\imath (\Omega_2 v_z - \Omega_3 v_y)
+ \hat\jmath (\Omega_3 v_x - \Omega_1 v_z)
+\hat{k} (\Omega_1 v_y - \Omega_2 v_x)}_{\vec\Omega\times\vec{v}} \, .
</math>
  </td>
</tr>
</table>

That is, along the way we derive a,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Hybrid (Eulerian) Representation'''</font><br />of the rotating-frame Euler Equation</td>
</tr>

<tr>
  <td align="right">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\vec\Omega \times \vec{v}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr] 
\, .
</math>
  </td>
</tr>
</table>

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

<div align="center"><font color="red"><b>CAUTION!</b></font></div>
If our interpretation of Chandrasekhar's discussion of "moving frames" is correct &#8212; see, Chap. 4, &sect;25 of  [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; the RHS of his Eq. (18) should match the LHS of our "hybrid" equation, but it does not:  the pair of vector velocities in his advection term are swapped.  That is, based on our interpretation, the RHS of his Eq. (18) reads,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v} \, ;</math>
  </td>
</tr>
</table>
and this expression carries over to the LHS of his Eq. (19).  This is either a mistake in his presentation, or our interpretation of his presentation is incorrect.
</td></tr></table>

==Coordinate Transformation==

===Traditional Presentation===
At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (''i.e.,'' time-independent) angular velocity <math>~\Omega_f</math>.  In order to transform any one of the [[PGE#Principal_Governing_Equations|principal governing equations]] from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>~d/dt</math> operator, which denotes Lagrangian time-differentiation in the inertial frame, must everywhere be replaced as follows:
<div align="center">
<math>
\biggl[\frac{d}{dt} \biggr]_{inertial} \rightarrow \biggl[\frac{d}{dt} \biggr]_{rot} + {\vec{\Omega}}_f \times .
</math>

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (11)

</div>

Operating on the fluid element's position vector, <math>\vec{x}</math>, we obtain the transformation,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\frac{d\vec{x}}{dt}\biggr|_\mathrm{inertial}</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="right"><math>\frac{d\vec{x}}{dt}\biggr|_\mathrm{rotating} + \vec\Omega \times \vec{x} \, ,</math></td>
</tr>
</table>
that is,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\vec{v}_\mathrm{inertial}</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="right"><math>\vec{v}_\mathrm{rot} + \vec\Omega \times \vec{x} \, .</math></td>
</tr>
<tr>
  <td align="center" colspan="3" align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (15)
  </td>
</tr>
</table>

Performing this transformation implies, for example, that
<div align="center">
<math>
\vec{v}_{inertial} = \vec{v}_{rot} + {\vec{\Omega}}_f \times \vec{x} ,
</math>
</div>
and,
<div align="center">
<math>
\biggl[ \frac{d\vec{v}}{dt}\biggr]_{inertial} = \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} + {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) 
</math>

<math>
= \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - \frac{1}{2} \nabla \biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] 
</math>
</div>
(If we were to allow <math>{\vec\Omega}_f</math> to be a function of time, an additional term involving the time-derivative of <math>{\vec\Omega}_f</math> also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in [[Appendix/References|BT87]].)  Note as well that the relationship between the fluid [[PGE/RotatingFrame#WikiVorticity|vorticity]] in the two frames is,
<div align="center">
<math>
[\vec\zeta]_{inertial} = [\vec\zeta]_{rot} + 2{\vec\Omega}_f .
</math>
</div>

===Chandrasekhar's Approach===

Here we draw extensively from Chapter 4, &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].

====Transformation Matrix====

Following Chandrasekhar, we let <math>\vec{X}</math> represent the inertial-frame position vector of a fluid element, in which case <math>d\vec{X}/dt</math> is the inertial-frame velocity <math>(\vec{v})</math> of that fluid element, and the acceleration, <math>d\vec{v}/dt</math>, that appears on the LHS of  the [[PGE/Euler#Lagrangian_Representation|Lagrangian representation of the (intertial-frame) Euler equation]] may be rewritten as the second time-derivative of <math>\vec{X}</math>, namely,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Lagrangian Representation'''</font><br />of the (inertial-frame) Euler Equation</td>
</tr>
<tr>
  <td align="right">
<math>\frac{d^2\vec{X}}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{inertial} \, .</math>
  </td>
</tr>
</table>

Chandrasekhar uses the matrix, <math>\mathbf{T}(t)</math>, to represent the (time-dependent) linear transformation that relates <math>\vec{X}</math> to  the corresponding moving-frame position vector, <math>\vec{x}</math>.  Specifically, he sets,

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

<tr>
  <td align="right">
<math>\vec{x}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\mathbf{T}\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, Eq. (1)</td>
</tr>
</table>
Applying the same transformation to the inertial-frame velocity, <math>d\vec{X}/dt</math>, gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\vec{U}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\mathbf{T}\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. (14a)</td>
</tr>
</table>
which Chandrasekhar refers to as the velocity in the <b>inertial frame</b> that has been <font color="darkgreen">"resolved along the instantaneous coordinate axes of the moving frame."</font>  And applying this transformation to the inertial-frame acceleration gives the term, <math>\mathbf{T} [d^2\vec{X}/dt^2]</math>, which Chandrasekhar describes as representing <font color="darkgreen">"&hellip; the acceleration in the <b>inertial frame</b> resolved, however, along the instantaneous directions of the coordinate axes of the moving frame."</font>  Applying the transformation to both sides of the Lagrangian representation of the Euler equation gives,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\mathbf{T} \frac{d^2\vec{X}}{dt^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{moving} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, combination of Eqs. (16) &amp; (17)</td>
</tr>
</table>
where, as Chandrasekhar clarifies, the gradients on the RHS must be <font color="darkgreen">"&hellip; evaluated in the coordinates of the moving frame."</font>

====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.

----

====Component Form====

<font color="red">EXAMPLE #1:</font>  Inertial-frame velocities, <math>\vec{U}</math>, as viewed in the inertial frame.

Adding the Euler equation,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\rho\frac{d\vec{U}}{dt}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\nabla P - \rho\nabla\Phi \, ,
</math>
  </td>
</tr>
</table>
to the product of the inertial-frame velocity and the equation of continuity,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\vec{U}~\frac{d\rho}{dt}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\rho\vec{U}(\nabla\cdot \vec{u})\, ,
</math>
  </td>
</tr>
</table>
gives,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\frac{d(\rho\vec{U})}{dt} + \rho\vec{U}(\nabla\cdot \vec{u})</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\nabla P - \rho\nabla\Phi 
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{\partial(\rho\vec{U})}{\partial t} + \nabla\cdot [(\rho\vec{U})\vec{u}]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-~\nabla P - \rho\nabla\Phi 
</math>
  </td>
</tr>
</table>


<span id="ComponentForm">In component form,</span> the relation between <math>\vec{U}</math> and <math>\vec{u}</math> reads,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="left">
<math> 
U_i
=
u_i - (\Omega^*)_{ik}x_k 
=
u_i + \overbrace{\epsilon_{ijk} \Omega_j x_k}^{[~\vec{\Omega} \times \vec{x}~]_i} 
\, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="1">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (21)</td>
</tr>
</table>

and the rotating-frame Euler equation becomes,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math> 
U_i
=
u_i - (\Omega^*)_{ik}x_k 
=
u_i + \overbrace{\epsilon_{ijk} \Omega_j x_k}^{[~\vec{\Omega} \times \vec{x}~]_i} 
\, .
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>

</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="1">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (21)</td>
</tr>
</table>

===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>

==Continuity Equation (rotating frame)==
Applying these transformations to the standard, inertial-frame representations of the continuity equation presented [[PGE/ConservingMass#Continuity_Equation|elsewhere]], we obtain the:

<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Continuity Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{d\rho}{dt} \biggr]_{rot} + \rho \nabla \cdot {\vec{v}}_{rot} = 0</math> ;
</div>


<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Continuity Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{\partial\rho}{\partial t} \biggr]_{rot} + \nabla \cdot (\rho {\vec{v}}_{rot}) = 0</math> .
</div>


==Euler Equation (rotating frame)==
Applying these transformations to the standard, inertial-frame representations of the Euler equation presented [[PGE/Euler#Euler_Equation|elsewhere]], we obtain the:

<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math> ;
</div>


<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math> ;
</div>


<div align="center">
Euler Equation<br />
written <font color="#770000">'''in terms of the Vorticity'''</font> and<br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}v_{rot}^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .
</div>


==Centrifugal and Coriolis Accelerations==

Following along the lines of the discussion presented in Appendix 1.D, &sect;3 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], in a rotating reference frame the Lagrangian representation of the Euler equation may be written in the form,
<div align="center">
<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi + {\vec{a}}_{fict} </math>,
</div>
where,
<div align="center">
<math>
{\vec{a}}_{fict} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) .
</math>
</div>
So, as viewed from a rotating frame of reference, material moves as if it were subject to two ''fictitious accelerations'' which traditionally are referred to as the,
<div align="center">
<font color="#770000">'''Coriolis Acceleration'''</font> 

<math>
{\vec{a}}_{Coriolis} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} ,
</math>
</div>

(see the related [[PGE/RotatingFrame#WikiCoriolis|Wikipedia discussion]]) and the
<div align="center">
<font color="#770000">'''Centrifugal Acceleration'''</font> 

<math>
{\vec{a}}_{Centrifugal} \equiv - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})
= \frac{1}{2} \nabla\biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2  \biggr] 
</math>
</div>
(see the related [[PGE/RotatingFrame#WikiCentrifugal|Wikipedia discussion]]).

==Nonlinear Velocity Cross-Product==

In some contexts &#8212; for example, our discussion of [[Apps/RiemannEllipsoidsCompressible|Riemann ellipsoids]] or the analysis by [[Apps/Korycansky_Papaloizou_1996|Korycansky &amp; Papaloizou (1996)]] of nonaxisymmetric disk structures &#8212; it proves useful to isolate and analyze the term in the "vorticity formulation" of the Euler equation that involves a nonlinear cross-product of the rotating-frame velocity vector, namely,
<div align="center">
<math>
\vec{A} \equiv ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot} .
</math>
</div>

NOTE: To simplify notation, for most of the remainder of this subsection we will drop the subscript "rot" on both the velocity and vorticity vectors.

===Align &Omega;<sub>f</sub> with z-axis===
Without loss of generality we can set <math>{\vec\Omega}_f = \hat{k}\Omega_f</math>, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system.    The Cartesian components of <math>{\vec{A}}</math> are then,
<div align="left">
<math>
\hat{i}: ~~~~~~ A_x = \zeta_y v_z - (\zeta_z + 2\Omega) v_y  ,
</math><br />

<math>
\hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x - \zeta_x v_z  ,
</math><br />

<math>
\hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x  ,
</math>
</div>
where it is understood that the three Cartesian components of the vorticity vector are,
<div align="center">
<math>
\zeta_x = \biggl[\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \biggr] ,
 ~~~~~~ \zeta_y = \biggl[ \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \biggr] ,
 ~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .
</math>
</div>
In turn, the curl of <math>\vec{A}</math> has the following three Cartesian components: 
<div align="left">
<math>
\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr],
</math><br />

<math>
\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial z}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,
</math><br />

<math>
\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr] - \frac{\partial}{\partial y}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] .
</math>
</div>

===When v<sub>z</sub> = 0 ===

If we restrict our discussion to configurations that exhibit only planar flows &#8212; that is, systems in which <math>v_z = 0</math> &#8212; then the Cartesian components of <math>{\vec{A}}</math> and <math>\nabla\times\vec{A}</math> simplify somewhat to give, respectively,
<div align="left">
<math>
\hat{i}: ~~~~~~ A_x = - (\zeta_z + 2\Omega) v_y  ,
</math><br />

<math>
\hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x  ,
</math><br />

<math>
\hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x  ,
</math>
</div>

and,
<div align="left">
<math>
\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x \biggr],
</math><br />

<math>
\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = - \frac{\partial}{\partial z}\biggl[(\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,
</math><br />

<math>
\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x  \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] ,
</math>
</div>
where, in this case, the three Cartesian components of the vorticity vector are,
<div align="center">
<math>
\zeta_x = - \frac{\partial v_y}{\partial z} ,
 ~~~~~~ \zeta_y = \frac{\partial v_x}{\partial z} ,
 ~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .
</math>
</div>

=Related Discussions=
* <span id="WikiVorticity">Wikipedia discussion of [http://en.wikipedia.org/wiki/Vorticity vorticity].</span>
* <span id="WikiCoriolis">Wikipedia discussion of [http://en.wikipedia.org/wiki/Coriolis_effect#Formula Coriolis Effect].</span>
* <span id="WikiCentrifugal">Wikipedia discussion of [http://en.wikipedia.org/wiki/Centrifugal_force#Derivation_using_vectors Centrifugal acceleration].</span>


<ul>
  <li>
An [[PGE/ConservingMomentum#Euler_Equation|earlier draft of the ''inertial-frame Euler equation'' presentation]].
  </li>
  <li>
Euler equation viewed from a ''rotating frame of reference''.
  <ul>
    <li>[[PGE/RotatingFrame|Revised Presentation]] which includes a relevant EFE discussion</li>
    <li>[[PGE/RotatingFrameOld|Initial Presentation]]</li>
    <li>[[Appendix/Ramblings/HybridSchemeOld|Hybrid Scheme]]</li>
  </ul>
  </li>
</ul>


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