Editing Appendix/Ramblings/NonlinarOscillation (section)

====Setup====

Now, let's identify two <math>n=5</math> equilibrium states that sit very near the <math>P_\mathrm{max}</math> turning point on the two separate branches of the equilibrium sequence and that have identical external pressures.  We know [[SSC/FreeEnergy/Powerpoint#Case_M_Equilibrium_Conditions|from separate discussions]] that, in both cases,

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

<tr>
  <td align="right">
<math>
\frac{P_\mathrm{e}}{P_\mathrm{norm}}
</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{2\cdot 3^3}{\pi}\biggr]^{3}
\tilde\xi^{12} \tilde\theta_n^{6}( - \tilde\theta' )^{6} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \biggl[ \frac{\pi}{2\cdot 3^3}\biggr]^{1/2}\biggl[\frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]^{1/6}</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
\tilde\xi^{2} \tilde\theta_n( - \tilde\theta' )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
3\ell^2 (1+\ell^2)^{-1/2} \frac{\ell}{\sqrt{3}} (1+\ell^2)^{-3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\sqrt{3}\ell^3  (1+\ell^2)^{-2}
</math>
  </td>
</tr>
</table>
</div>
We can therefore write,

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

<tr>
  <td align="right">
<math>(1+\ell^2)^{2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
p_0\ell^3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~\ell^4 - p_0\ell^3 + 2\ell^2 + 1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>p_0 \equiv \biggl[ \frac{2\cdot 3^4}{\pi}\biggr]^{1/2}\biggl[\frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]^{-1/6}</math>
</div>

So, in essence, we seek two real roots of this quartic equation that are near <math>P_\mathrm{max}</math>, that is, that are near <math>\ell = \sqrt{3}</math> &#8212; where <math>p_0 = (2^8/3^3)^{1/2}</math>.


Because we are hunting for equilibrium configurations near <math>P_\mathrm{max}</math>, it makes sense to make the variable substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\ell</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>\rightarrow</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>\sqrt{3}(1+\epsilon) \, ,</math>
  </td>
</tr>
</table>
</div>
and look for pairs of values, <math>~\epsilon_\pm</math> (both real, but one positive and the other negative).

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>3^2(1+\epsilon)^4 - 3^{3/2}p_0(1+\epsilon)^3 + 6(1+\epsilon)^2 + 1</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
3^2\biggl[1 + 4\epsilon + 6\epsilon^2 + 4\epsilon^3 + \epsilon^4\biggr] 
- 3^{3/2}p_0\biggl[ 1+ 3\epsilon + 3\epsilon^2 + \epsilon^3  )\biggr] 
+ 6\biggl[1 + 2\epsilon + \epsilon^2\biggr] 
+ 1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
\epsilon^4 \biggl[ 9 \biggr]
+\epsilon^3 \biggl[ 36 - 3^{3/2}p_0 \biggr]
+\epsilon^2 \biggl[ 54 - 3^{5/2}p_0 + 6 \biggr]
+\epsilon \biggl[36 - 3^{5/2}p_0 + 12 \biggr]
+(16- 3^{3/2}p_0)\, .
</math>
  </td>
</tr>
</table>
</div>
And, because we will only be examining values of the external pressure that are ''less'' than <math>P_\mathrm{max}</math>, and we know that ''at'' the point of maximum pressure, <math>3^{3/2}p_0 = 16</math>, it makes sense to make the substitution,
<div align="center">
<math>3^{3/2}p_0  ~~~\rightarrow ~~~ (16+\delta) \, .</math>
</div>

Hence, for a fixed choice of <math>\delta</math> (reasonably small, and positive), we seek two real roots (one positive and the other negative) of the quartic relation,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
9\epsilon^4 
+\epsilon^3 \biggl[ 36 - (16+\delta ) \biggr]
+\epsilon^2 \biggl[ 60 - 3(16+\delta ) \biggr]
+\epsilon \biggl[48 - 3(16+\delta ) \biggr]
-\delta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
9\epsilon^4 
+\epsilon^3 (20-\delta  )
+\epsilon^2 ( 12 - 3\delta  )
-\epsilon (3\delta )
-\delta \, .
</math>
  </td>
</tr>
</table>
</div>

What are the reasonable limits on <math>~\delta</math>?  Well, first note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>p_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{(1+\ell^2)^2}{\ell^3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 16+\delta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^{3/2}\biggl[\frac{(1+\ell^2)^2}{\ell^3}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \delta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^{3/2}\biggl[\frac{(1+\ell^2)^2}{\ell^3}\biggr] - 2^4 \, .</math>
  </td>
</tr>
</table>
</div>
Now, according to our [[SSC/FreeEnergy/PowerPoint#Case_M_Equilibrium_Conditions|accompanying discussion]], the relevant limits on <math>\ell</math> are <math>\sqrt{3}</math> (set by the maximum pressure turning point) and 2.223175 (set by the transition to dynamical instability).  The corresponding values of <math>\delta</math> are: &nbsp; 0 (by design) and 0.69938.