Editing Aps/MaclaurinSpheroidFreeFall (section)

==Simplified Governing Relations==

When studying the dynamical evolution of strictly axisymmetric configurations, it proves useful to write the spatial operators in our overarching set of [[PGE|principal governing equations]] in terms of cylindrical coordinates, <math>(\varpi, \phi, z)</math>, and to simplify the individual equations as described in our [[AxisymmetricConfigurations/PGE#Governing_Equations_.28CYL..29|accompanying discussion]].  The resulting set of simplified governing relations is &hellip;

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} + \frac{\rho}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \varpi \dot\varpi \biggr] 
+ \rho \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math><br />


<span id="PGE:Euler">
<font color="#770000">'''Euler Equation'''</font>
</span><br />

<math>
{\hat{e}}_\varpi \biggl[ \frac{d \dot\varpi}{dt} -  \frac{j^2}{\varpi^3} \biggr]  + {\hat{e}}_z \biggl[ \frac{d \dot{z}}{dt} \biggr] = -
{\hat{e}}_\varpi \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] -  {\hat{e}}_z \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math><br />
where, the specific angular momentum, <math>j(\varpi,z) \equiv \varpi^2 \dot\phi =  \mathrm{constant} ~(\mathrm{i.e.,}~\mathrm{independent~of~time})
</math><br />



<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{Math/EQ_FirstLaw02}}


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho . 
</math><br />
</div>

This study is closely tied to our [[SSC/Dynamics/FreeFall#Free-Fall|separate discussion of the free-fall collapse of uniform-density ''spheres'']].  For example, by definition, an element of fluid is in "free fall" if its motion in a gravitational field is unimpeded by pressure gradients.  The most straightforward way to illustrate how such a system evolves is to set <math>P = 0</math> in all of the governing equations.  In doing this, the continuity equation and the Poisson equation remain unchanged; the equation formulated by the first law of thermodynamics becomes irrelevant; and the two components of the Euler equation become,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\hat{\mathbf{e}}_\varpi</math>:</td>
  <td align="right">
<math>\frac{d\dot{\varpi}}{dt} - \frac{j^2}{\varpi^3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\partial\Phi}{\partial\varpi}    \, ,</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\hat{\mathbf{e}}_z</math>:</td>
  <td align="right">
<math>\frac{d\dot{z}}{dt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\partial\Phi}{\partial z}    \, .</math>
  </td>
</tr>
</table>