Editing Aps/MaclaurinSpheroidFreeFall (section)

===Motion of a Single Particle===
Consider a particle that, at time <math>t=0</math>, is at position <math>(\varpi_0, \phi_0, z_0)</math> and is moving about the <math>z</math>-axis with velocity, <math>\varpi_0\Omega ~\Rightarrow~ j_0 = \varpi_0^2 \Omega</math>.  Consider furthermore that its acceleration is subject to the force arising from an axisymmetric gravitational potential of the form,

<div align="center">
<math>
\Phi(\vec{x}, t) = A(t) \varpi^2 + C(t) z^2 .
</math>
</div>

[This is the gravitational potential adopted by {{ LB62hereafter }} &#8212; see his equation (1) &#8212; except he adopted a different sign convention to ours.  He would therefore have also attached a sign to the gradient of the potential that is the opposite of the sign that appears on the right-hand side of our Euler equation expression.]  In this case, the two components of the Euler equation that govern the particle's motion are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\hat{\mathbf{e}}_\varpi</math>:</td>
  <td align="right">
<math>\ddot\varpi - \frac{j_0^2}{\varpi^3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2A\varpi    \, ,</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\hat{\mathbf{e}}_z</math>:</td>
  <td align="right">
<math>\ddot{z} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2Cz    \, ,</math>
  </td>
</tr>
</table>
where we have adopted the familiar shorthand notation, <math>d\dot\varpi/dt \rightarrow \ddot\varpi</math> and <math>d\dot{z}/dt \rightarrow \ddot{z}</math>.  If we divide the first of these relations by <math>\varpi_0</math> and the second by <math>z_0</math>, then adopt the dimensionless variables, <math>R \equiv \varpi/\varpi_0</math> and <math>Z \equiv z/z_0</math>, we can write,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\frac{\ddot\varpi}{\varpi_0}  - \frac{j_0^2}{\varpi_0 \varpi^3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align=left">
<math>- 2A \biggl( \frac{\varpi}{\varpi_0} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ddot{R}  - \frac{\Omega^2}{R^3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align=left">
<math>- 2A R \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LB62hereafter }}, p. 710, Eq. (10)
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\ddot{z}}{z_0} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2C \biggl(\frac{z}{z_0}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ddot{Z} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2C Z \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LB62hereafter }}, p. 710, Eq. (11)
  </td>
</tr>
</table>
Finally, we use <math>\varphi(t)</math> to represent the particle's time-varying angular-coordinate position ''relative to'' its initial position &#8212; that is, we adopt the definition, <math>\varphi(t) \equiv \phi(t) - \phi_0</math>. Then, conservation of angular momentum implies that, at any moment, the particle's rotation frequency about the symmetry axis will be,
<table border="0" cellpadding="0" align="center">

<tr>
  <td align="right">
<math>\dot\varphi = \dot\phi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\Omega}{R^2} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LB62hereafter }}, p. 710, Eq. (9)
  </td>
</tr>
</table>
This governing set of evolutionary equations has been set up such that at time, <math>t=0</math>: <math>R = 1</math>, <math>Z = 1</math>, <math>\varphi = 0</math>, <math>\dot{R} = 0</math>, <math>\dot{Z} = 0</math>, and <math>\dot\varphi = \Omega</math>. With this set of initial conditions in hand, along with an appropriate specification of the two time-dependent coefficients, <math>A(t)</math> and <math>C(t)</math>, the set of governing relations can be integrated (numerically) to give <math>R(t), \varphi(t)</math>, and <math>Z(t)</math>.  This is the result that {{ LB62 }} established for the motion of one particle.