Editing SSC/Structure/BiPolytropes/Analytic51Renormalize (section)

===2<sup>nd</sup> Trial===

The relevant LAWE for the envelope is,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl[ \frac{A\sin(\eta-B)}{\eta}\biggr]\frac{d^2x}{d\eta^2} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{2A}{\eta}\biggl\{ \sin(\eta - B) + \eta\cos(\eta - B) \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ \frac{2A}{\eta} \biggl\{ \sin(\eta - B) - \eta\cos(\eta - B) \biggr\}  \frac{x}{\eta^2}
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) x 
</math>
</td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{1}{3\beta} \cdot \frac{d^2x}{d\eta^2} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{2}{\eta}\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \biggl[ \frac{1}{3\beta } \cdot \frac{dx}{d\eta} \biggr] 
+ 2 \biggl\{ 1 - \eta\cot(\eta - B) \biggr\}  \frac{x}{3\beta\eta^2}
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) \biggl[ \frac{\eta}{3 \beta A\sin(\eta-B)}\biggr] x
\, . 
</math>
</td>
</tr>
</table>
Here, we will ''guess'' a displacement function, <math>x_\mathrm{env}</math>, of the form,

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

<tr>
  <td align="right">
<math>\frac{x_\mathrm{env}}{3\beta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta^2 } \biggl[1 - \eta\cot(\eta - B_x) \biggr]
=
\frac{1}{\eta^2 }  - \frac{\cos(\eta - B_x)}{\eta\sin(\eta - B_x)} 
\, , 
</math>
  </td>
</tr>
</table>
where we will assume, quite generally, that <math>B_x \ne B</math>.  The first and second derivatives of <math>x_\mathrm{env}</math> are,

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

<tr>
  <td align="right">
<math>\frac{1}{3\beta} \biggl[ \frac{dx_\mathrm{env}}{d\eta} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{2}{\eta^3 }  
+ 
\frac{\cos(\eta - B_x)}{\eta^2\sin(\eta - B_x)} 
+ 
\frac{\sin(\eta - B_x)}{\eta\sin(\eta - B_x)} 
+ 
\frac{\cos^2(\eta - B_x)}{\eta\sin^2(\eta - B_x)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{2}{\eta^3 }  
+ 
\frac{1}{\eta} 
+ 
\frac{\cos(\eta - B_x)}{\eta^2\sin(\eta - B_x)} 
+ 
\frac{\cos^2(\eta - B_x)}{\eta\sin^2(\eta - B_x)}
\, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{1}{3\beta} \biggl[ \frac{d^2x_\mathrm{env}}{d\eta^2} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{1}{\eta^2}
- \frac{2\cos(\eta-B_x)}{\eta^3\sin(\eta-B_x)}
+ \frac{1}{\eta^2}\biggl[
- \frac{\sin(\eta - B_x)}{\sin(\eta - B_x)}
-
\frac{\cos^2(\eta - B_x)}{\sin^2(\eta - B_x)}
\biggr]
- 
\frac{\cos^2(\eta - B_x)}{\eta^2\sin^2(\eta - B_x)}
+ \frac{1}{\eta} \biggl[
-
\frac{2\cos(\eta - B_x)}{\sin(\eta - B_x)}
-
\frac{2\cos^3(\eta - B_x)}{\sin^3(\eta - B_x)}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{1}{\eta^2}
- \frac{2\cos(\eta-B_x)}{\eta^3\sin(\eta-B_x)}
- \frac{1}{\eta^2}\biggl[
1
+
\frac{\cos^2(\eta - B_x)}{\sin^2(\eta - B_x)}
\biggr]
- 
\frac{\cos^2(\eta - B_x)}{\eta^2\sin^2(\eta - B_x)}
- \frac{1}{\eta}\biggl[
\frac{2\cos(\eta - B_x)}{\sin(\eta - B_x)}
+
\frac{2\cos^3(\eta - B_x)}{\sin^3(\eta - B_x)}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{2}{\eta^2}
- \frac{2\cos(\eta-B_x)}{\eta^3\sin(\eta-B_x)}
- \frac{2}{\eta^2}\biggl[
\frac{\cos^2(\eta - B_x)}{\sin^2(\eta - B_x)}
\biggr]
- \frac{2}{\eta}\biggl[
\frac{\cos(\eta - B_x)}{\sin(\eta - B_x)}
+
\frac{\cos^3(\eta - B_x)}{\sin^3(\eta - B_x)}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{2}{\eta^2}\biggl[1 + \cot^2(\eta - B_x) \biggr]
- \frac{2\cot(\eta-B_x)}{\eta^3}
- \frac{2}{\eta}\biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]\, .
</math>
  </td>
</tr>
</table>
These match the expressions for <math>dx_P/d\eta</math> and <math>d^2x_P/d\eta^2</math> that we separately derived in a [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B|subsection labeled ''Attempt_4B'' of an accompanying discussion labeled]].  Plugging these three relations into the LAWE, then multiplying through by <math>\eta^4</math>, gives,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>
\frac{6}{\eta^4} - \frac{2}{\eta^2}\biggl[1 + \cot^2(\eta - B_x) \biggr]
- \frac{2\cot(\eta-B_x)}{\eta^3}
- \frac{2}{\eta}\biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]
</math>
</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{2}{\eta}\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \biggl[ -\frac{2}{\eta^3 }  
+ 
\frac{1}{\eta} 
+ 
\frac{\cot(\eta - B_x)}{\eta^2} 
+ 
\frac{\cot^2(\eta - B_x)}{\eta}
 \biggr] 
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
2 \biggl[ 1 - \eta\cot(\eta - B) \biggr]  \frac{1}{\eta^2}
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) \biggl[ \frac{\eta}{ A\sin(\eta-B)}\biggr]
\biggr\} \biggl[ \frac{1}{\eta^2 }  - \frac{\cot(\eta - B_x)}{\eta}\biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
6 - 2\eta^2 \biggl[1 + \cot^2(\eta - B_x) \biggr]
- 2\eta \cot(\eta-B_x)
- 2\eta^3 \biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]
</math>
</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \biggl[ -2  
+ 
\eta^2 
+ 
\eta\cot(\eta - B_x) 
+ 
\eta^2 \cot^2(\eta - B_x)
 \biggr] 
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
2 \biggl[ 1 - \eta\cot(\eta - B) \biggr] 
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) \biggl[ \frac{\eta^3}{ A\sin(\eta-B)}\biggr]
\biggr\} \biggl[ 1  - \eta \cot(\eta - B_x)\biggr]
</math>
</td>
</tr>
</table>

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl( \frac{\sigma_c^2}{6} \biggr) \biggl[ \frac{\eta^3}{ A\sin(\eta-B)}\biggr]
\biggl[ 1  - \eta \cot(\eta - B_x)\biggr]
</math>
</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 1 + \eta\cot(\eta - B) \biggr] \biggl[ 4  
- 
2\eta^2 
- 
2\eta\cot(\eta - B_x) 
- 
2\eta^2 \cot^2(\eta - B_x)
 \biggr] 
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 
2 \biggl[ 1 - \eta\cot(\eta - B) \biggr] 
\biggl[ 1  - \eta \cot(\eta - B_x)\biggr]

-6 + 2\eta^2 \biggl[1 + \cot^2(\eta - B_x) \biggr]
+ 2\eta \cot(\eta-B_x)
+ 2\eta^3 \biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 4 - 2\eta^2 - 2\eta\cot(\eta - B_x) - 2\eta^2 \cot^2(\eta - B_x)\biggr] 
+
\eta\cot(\eta - B) 
\biggl[ 4 - 2\eta^2 - 2\eta\cot(\eta - B_x) - 2\eta^2 \cot^2(\eta - B_x)\biggr] 

</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 
\biggl[ 2 - 2\eta\cot(\eta - B) - 2\eta \cot(\eta - B_x) + 2\eta^2 \cot(\eta - B)\cot(\eta - B_x)\biggr]

-6 + 2\eta^2 + 2\eta^2 \cot^2(\eta - B_x) 
+ 2\eta \cot(\eta-B_x)
+ 2\eta^3 \cot(\eta - B_x) 
+ 2\eta^3 \cot^3(\eta - B_x)
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\eta\cot(\eta - B_x) - 2\eta^2 \cot^2(\eta - B_x) 
+
4\eta\cot(\eta - B) - 2\eta^3\cot(\eta - B) - 2\eta^2 \cot(\eta - B)\cot(\eta - B_x) - 2\eta^3\cot(\eta - B) \cot^2(\eta - B_x)

</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- 2\eta\cot(\eta - B) - 2\eta \cot(\eta - B_x) + 2\eta^2 \cot(\eta - B)\cot(\eta - B_x)
+ 2\eta^2 \cot^2(\eta - B_x) 
+ 2\eta \cot(\eta-B_x)
+ 2\eta^3 \cot(\eta - B_x) 
+ 2\eta^3 \cot^3(\eta - B_x)
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\eta^3 \biggl[\cot(\eta - B)  +\cot(\eta - B) \cot^2(\eta - B_x)
- \cot(\eta - B_x) - \cot^3(\eta - B_x) \biggr]
+ 2\eta \biggl[\cot(\eta - B) - \cot(\eta - B_x) \biggr]
</math>
</td>
</tr>
</table>

Notice that if we set <math>B_x = B</math>, the RHS of this LAWE expression goes to zero.  This [thankfully] is as expected.