Editing PGE/Euler (section)

==Lagrangian Representation==

===in terms of velocity:===
Among the [[PGE#Principal_Governing_Equations|principal governing equations]] we have included the
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<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the Euler Equation,

{{Template:Math/EQ_Euler01}}

[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.55)
</div>

===in terms of momentum density:===
<!--
Multiplying this equation through by the mass density {{Template:Math/VAR_Density01}} produces the relation,
<div align="center">
<math>\rho\frac{d\vec{v}}{dt} = - \nabla P - \rho\nabla \Phi</math> ,
</div>
which may be rewritten as,
<div align="center">
<math>\frac{d(\rho\vec{v})}{dt}- \vec{v}\frac{d\rho}{dt} = - \nabla P - \rho\nabla \Phi</math> .
</div>
Combining this with the [[PGE/ConservingMass#ConservingMass:Lagrangian|Standard Lagrangian Representation of the Continuity Equation]], we derive,
<div align="center">
<math>\frac{d(\rho\vec{v})}{dt}+ (\rho\vec{v})\nabla\cdot\vec{v} = - \nabla P - \rho\nabla \Phi</math> .
</div>
-->

Multiplying this equation through by the mass density {{Template:Math/VAR_Density01}} produces the relation,
<table border="0" align="center" cellpadding="5"><tr><td align="left">
<math>\rho\frac{d\vec{v}}{dt} = - \nabla P - \rho\nabla \Phi</math> ,
</td></tr></table>

which may be rewritten as,

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{d(\rho\vec{v})}{dt}- \vec{v}\frac{d\rho}{dt} </math>
  </td>
  <td align="center">
<math>=</math> 
  </td>
  <td align="left">
<math>- \nabla P - \rho\nabla \Phi</math> .
  </td>
</tr>
</table>

Combining this with the [[PGE/ConservingMass#ConservingMass:Lagrangian|Standard Lagrangian Representation of the Continuity Equation]], we derive,
<table border="0" align="center" cellpadding="5"><tr><td align="left">
<math>\frac{d(\rho\vec{v})}{dt}+ (\rho\vec{v})\nabla\cdot\vec{v} = - \nabla P - \rho\nabla \Phi</math> .
</td></tr></table>