Editing ThreeDimensionalConfigurations/Stability/RiemannEllipsoids (section)

==Self-Adjoint Sequences==

What are the expressions that define the upper <math>(x = -1)</math> and lower <math>(x = +1)</math> boundaries of the ''horned shaped'' region of equilibrium S-Type Riemann Ellipsoids? Well, as we have [[ThreeDimensionalConfigurations/RiemannStype#Based_on_Virial_Equilibrium|discussed in an associated chapter]], the value of the parameter, <math>x</math>, that is associated with each point <math>(b/a, c/a)</math> within the ''horned shaped'' region is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
1 +2Cx + x^2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">{{ LL96 }}, &sect;2, Eq. (5)</td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>C</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a b B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]  \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">{{ LL96 }}, &sect;2, Eq. (6)</td></tr>
</table>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~A_{12}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (107)</font> ]</td></tr>

<tr>
  <td align="right">
<math>~B_{12}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~A_2 - a^2A_{12} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (105)</font> ]<br />See also the ''note'' immediately following &sect;21, Eq. (127)</td></tr>
</table>

===Upper Boundary===
The upper boundary of the ''horn-shaped'' region is obtained by setting <math>x = -1</math>.  That is, it is associated with coordinate pairs <math>(b/a, c/a)</math> for which,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
1 - 2C + 1
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ C
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>+1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[ \frac{a b B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>+1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ a b B_{12} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c^2 A_3 - a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~c^2 A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a b [ A_2 - a^2A_{12}]  + a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a b A_2  + b a^2 A_{12} (b  - a)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a b A_2  + b a^2 (a - b )\biggl[\frac{A_1-A_2}{a^2 - b^2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_2  + a \biggl[\frac{A_1-A_2}{a+b} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2(a+b)}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>aA_1 + bA_2 \, .</math>
  </td>
</tr>
</table>
Now, from the [[ThreeDimensionalConfigurations/RiemannStype#Riemann_S-Type_Ellipsoids|expressions for A<sub>1</sub>, A<sub>2</sub>, and A<sub>3</sub>]], we can furthermore write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
c^2(a+b) A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b A_1 + ab^2 [2 -(A_1 + A_3)] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b A_1 + 2ab^2 - ab^2 A_1 - ab^2 A_3 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ c^2(a+b) A_3 + ab^2 A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 2ab^2 + a^2 b A_1 - ab^2 A_1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{a}{b}\biggl[c^2(a+b)+ ab^2 \biggr]A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 2a^2b + a^2(a  - b)A_1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[c^2(a+b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> a^2b + bc(a  - b)
\biggl\{
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr]
\biggr\} \, ,</math>
  </td>
</tr>
</table>
where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;17, Eq. (32)</font> ]</td></tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">STRATEGY</font> for finding the locus of points that define the upper boundary of the horned-shape region &hellip; &nbsp; &nbsp;Set <math>a = 1</math>, and pick a value for <math>0 < b < 1</math>; then, using an iterative technique, vary <math>c</math> until the following expression is satisfied:

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

<tr>
  <td align="right">
<math>
\biggl[c^2(a+b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> a^2b + bc(a  - b)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr]
\, .</math>
  </td>
</tr>
</table>
Choose another value of <math>0 < b < 1</math>, then iterate again to find the value of <math>c</math> that corresponds to this new, chosen value of <math>b</math>.  Repeat!
</td></tr></table>

===Lower Boundary===

Similarly, the lower boundary is obtained by setting <math>x = +1</math>, that is, it is associated with coordinate pairs <math>(b/a, c/a)</math> for which,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
C
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[ \frac{a b B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~- a b B_{12} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c^2 A_3 - a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~c^2 A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- a b [ A_2 - a^2A_{12}]  + a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-a b A_2  + b a^2 A_{12} (b  + a)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-a b A_2  - b a^2 (a + b )\biggl[\frac{A_1-A_2}{a^2 - b^2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-A_2  - a \biggl[\frac{A_1-A_2}{a-b} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2(a-b)}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_2(b-a) - aA_1 + aA_2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>bA_2 - aA_1 \, .</math>
  </td>
</tr>
</table>

Now, from the [[ThreeDimensionalConfigurations/RiemannStype#Riemann_S-Type_Ellipsoids|expressions for A<sub>1</sub>, A<sub>2</sub>, and A<sub>3</sub>]], we can furthermore write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
c^2(a-b) A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2ab^2  - ab^2 A_1 - ab^2 A_3 - a^2 b A_1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ c^2(a-b) A_3  + ab^2 A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2ab^2 - ab(b + a)A_1  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{a}{b}\biggl[ c^2(a-b)+ ab^2 \biggr]A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2\biggl[ 2b - (b + a)A_1 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[ c^2(a-b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b - bc(b + a) 
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">STRATEGY</font> for finding the locus of points that define the lower boundary of the horned-shape region &hellip; &nbsp; &nbsp;Set <math>a = 1</math>, and pick a value for <math>0 < b < 1</math>; then, using an iterative technique, vary <math>c</math> until the following expression is satisfied:

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

<tr>
  <td align="right">
<math>
\biggl[ c^2(a-b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b - bc(b + a) 
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, .
</math>
  </td>
</tr>
</table>
Choose another value of <math>0 < b < 1</math>, then iterate again to find the value of <math>c</math> that corresponds to this new, chosen value of <math>b</math>.  Repeat!
</td></tr></table>