Editing ThreeDimensionalConfigurations/Stability/RiemannEllipsoids (section)

===Euler Equation===
From our initial overarching presentation of the principal governing equation, we draw an expression for the,

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<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 
~\underbrace{- ~2{\vec{\Omega}}_f \times {\vec{v}}_{rot}}_\mathrm{Coriolis} 
~\underbrace{- ~{\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})}_\mathrm{Centrifugal} 
\, .</math>
</div>

Moving the term that accounts for the Coriolis acceleration to the left-hand side of this expression, and realizing that the centrifugal acceleration may be rewritten in the form,
<div align="center">
<font color="#770000">'''Centrifugal Acceleration'''</font> 

<math>
{\vec{a}}_\mathrm{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>
the Euler equation becomes,
<table border="0" cellpadding="8" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> - \frac{1}{\rho} \nabla P - \nabla \Phi 
+ ~\frac{1}{2} \nabla\biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2  \biggr] 
\, .</math>
  </td>
</tr>
</table>
Except for the adopted sign convention for the gravitational potential, <math>\Phi \leftrightarrow -\Phi_\mathrm{L89}</math>, this precisely matches Equation (2) of {{ Lebovitz89ahereafter }}, namely,
<div align="center" id="EulerRotating">
<table border="1" align="center" cellpadding="8" width="80%">
<tr><td align="center" bgcolor="lightgreen">{{ Lebovitz89afigure }}</td></tr>
<tr><td align="left">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\frac{D\mathbf{u}}{Dt} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ \rho^{-1} \nabla p + \mathbf\nabla \{ \Phi_\mathrm{L89} + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89a }}, §2, p. 223, Eq. (2)<br />
{{ LL96b }}, &sect;2, p. 929, Eq. (2.1)
  </td>
</tr>
</table>
</td></tr>
</table>
</div>

In what follows, we will adopt the {{ Lebovitz89ahereafter }} variable notation.