Editing ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI

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=Description of Riemann Type I Ellipsoids=
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! style="height: 125px; width: 125px; background-color:#ffeeee;" |[[H_BookTiledMenu#Three-Dimensional_Configurations|<b>Type I<br />Riemann<br />Ellipsoids</b>]]
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A more succinct version of this chapter is titled, [[3Dconfigurations/DescriptionOfRiemannTypeI|3Dconfigurations/DescriptionOfRiemannTypeI]].
==Analytic Determination of Equilibrium Model Parameters==
Drawing heavily from &sect;47 (pp. 129 - 132) of [[Appendix/References#EFE|[<font color="red">EFE</font>] ]], in a separate chapter we show how the steady-state 2<sup>nd</sup>-order tensor virial equations can be used to derive the equilibrium structure of [[VE/RiemannEllipsoids#Riemann_Ellipsoids_of_Types_I,_II,_&_III|Riemann Ellipsoids of Type I, II, &amp; III]].  By definition, for these types of Riemann Ellipsoids, the two vectors <math>\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>y-z</math> plane &#8212; that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>.  For a given specified density <math>(\rho)</math> and choice of the three semi-axes <math>(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in that chapter's [[VE/RiemannEllipsoids#SummaryTable|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns:  <math>\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>.  

In an effort to simplify the constraint-equation expressions &#8212; and following the notation found in [[Appendix/References#EFE|[<font color="red">EFE</font>] ]] &#8212; we adopt the intermediary parameters:
<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\beta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{4a^2}\biggl\{ (4a^2 -b^2 + c^2  ) \mp \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, ;
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (16)</font> ]</td></tr>

<tr>
  <td align="right">
<math>
\gamma 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{4a^2}\biggl\{ (4a^2  + b^2 -c^2)  \mp \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (17)</font> ]</td></tr>
</table>

As is emphasized in [[Appendix/References#EFE|EFE]] (Chapter 7, &sect;47, p. 131) "<font color="darkgreen">&hellip; the signs in front of the radicals, in the two expressions, go together.</font>  Furthermore, "<font color="darkgreen">the two roots &hellip; correspond to the fact that, consistent with Dedekind's theorem, two states of internal motions are compatible with the same external figure.</font>"
</td></tr></table>


The five relevant constraint equations are,

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

<tr>
  <td align="right">
<math>2\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi G \rho c^2 \biggl\{ A_3  
+~
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  } \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>0
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho 
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{a^2b^2} \biggr] \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (170)</font> ]</td></tr>
</table>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\zeta_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_2 \beta \biggl[ \frac{a^2 + c^2}{c^2} \biggr] \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (12)</font> ]</td></tr>
</table>

<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>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho 
\biggl[ \frac{ a^2( b^2 + 3c^2 - 4a^2 ) B_{23} +c^2(a^2A_1 - b^2 A_2)  }{a^2 c^2 }
\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (171)</font> ]</td></tr>
</table>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\zeta_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3 \gamma \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (12)</font> ]</td></tr>
</table>

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

<tr>
  <td align="right">
<math>B_{23}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;21, Eqs. (105) &amp; (107)</font> ]</td></tr>
</table>

Throughout the [[Appendix/References#EFE|EFE]] discussion  of Type I Riemann Ellipsoids, the assumption is made that <math>b \ge a \ge c</math>.  Presumably then, when evaluating the above set of constraints, we must [[VE/RiemannEllipsoids#Specific_Case_of_a2_%3E_a1_%3E_a3|adopt the associations]], <math>(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>(A_3, a_3) \leftrightarrow (A_s, a_s)</math>.  This means that the coefficients, <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
2\biggl( \frac{a_1}{a_2}\biggr) \biggl[  \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_1 = 2 - (A_2 + A_3)
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{ 2a_1 a_3}{a_2^2 }
\biggl[ \frac{
E(\theta, k) 
-~(1-k^2)
F(\theta, k)
-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta = \cos^{-1} \biggl(\frac{a_3}{a_2} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>k = \biggl[\frac{1 - (a_1/a_2)^2}{1 - (a_3/a_2)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>

==Example Equilibrium Models==

===Extracted from XXVIII===
Here are equilibrium model parameters drawn from {{ Chandrasekhar66_XXVIIIfull }}.

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="16" bgcolor="lightgreen">
'''Data Extracted from Table 4 (p. 858) of <br />
{{ Chandrasekhar66_XXVIIIfigure }}
</td>
<td align="center" colspan="3" rowspan="2">Our<br /> (reverse-engineered)<br /> Determination</td>
</tr>
<tr>
  <td align="center" colspan="16">''The Properties of a Few Riemann Ellipsoids of Type I''</td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>\frac{a_3}{a_1}</math></td>
  <td align="center" rowspan="10" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="6">Direct</td>
  <td align="center" rowspan="10" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="6">Adjoint</td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>1</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>2</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>3</sub></font></td>
</tr>
<tr>
  <td align="center"><math>\Omega_2</math></td>
  <td align="center"><math>\Omega_3</math></td>
  <td align="center"><math>\zeta_2</math></td>
  <td align="center"><math>\zeta_3</math></td>

  <td align="center"><math>(\zeta_2/\Omega_2)</math></td>
  <td align="center"><math>(\zeta_3/\Omega_3)</math></td>

  <td align="center"><math>\Omega_2^\dagger</math></td>
  <td align="center"><math>\Omega_3^\dagger</math></td>
  <td align="center"><math>\zeta_2^\dagger</math></td>
  <td align="center"><math>\zeta_3^\dagger</math></td>

  <td align="center"><math>(\zeta_2/\Omega_2)^\dagger</math></td>
  <td align="center"><math>(\zeta_3/\Omega_3)^\dagger</math></td>
</tr>
<tr>
  <td align="right">1.05263</td>
  <td align="right">0.41667</td>
  <td align="right">+ 0.14834</td>
  <td align="right">+0.73257</td>
  <td align="right">-1.41355</td>
  <td align="right">-2.61578</td>

  <td align="right">-9.52912</td>
  <td align="right">-3.57069</td>

  <td align="right">+0.50185</td>
  <td align="right">+1.30617</td>
  <td align="right">-0.41783</td>
  <td align="right">-1.46707</td>
  
  <td align="right">-0.83258</td>
  <td align="right">-1.12318</td>

  <td align="right">0.43008706</td>
  <td align="right">0.40190235</td>
  <td align="right">1.16801059</td>
</tr>
<tr>
  <td align="right">1.25000</td>
  <td align="right">0.50000</td>
  <td align="right">+0.39259</td>
  <td align="right">+0.66536</td>
  <td align="right">-2.19983</td>
  <td align="right">-1.93895</td>

  <td align="right" bgcolor="yellow">-5.60338</td>
  <td align="right" bgcolor="yellow">-2.91414</td>

  <td align="right">+0.87993</td>
  <td align="right">+0.94583</td>
  <td align="right">-0.98148</td>
  <td align="right">-1.36398</td>
  
  <td align="right" bgcolor="pink">-1.11541</td>
  <td align="right" bgcolor="pink">-1.44210</td>

  <td align="right">0.50823343</td>
  <td align="right">0.37944073</td>
  <td align="right">1.11232585</td>
</tr>
<tr>
  <td align="right">1.44065</td>
  <td align="right">0.49273</td>
  <td align="right">+0.57179</td>
  <td align="right">+0.59896</td>
  <td align="right">-2.24560</td>
  <td align="right">-1.49425</td>

  <td align="right">-3.92732</td>
  <td align="right">-2.49474</td>

  <td align="right">+0.89032</td>
  <td align="right">+0.69996</td>
  <td align="right">-1.44219</td>
  <td align="right">-1.27866</td>

  <td align="right">-1.61986</td>
  <td align="right">-1.82676</td>

  <td align="right">0.52403947</td>
  <td align="right">0.32351421</td>
  <td align="right">1.15244632</td>
</tr>
<tr>
  <td align="right">1.66667</td>
  <td align="right">0.33333</td>
  <td align="right">+0.71251</td>
  <td align="right">+0.52815</td>
  <td align="right">-2.37502</td>
  <td align="right">-1.19714</td>

  <td align="right">-3.33331</td>
  <td align="right">-2.26667</td>

  <td align="right">+0.71251</td>
  <td align="right">+0.52815</td>
  <td align="right">-2.37502</td>
  <td align="right">-1.19714</td>

  <td align="right">-3.33331</td>
  <td align="right">-2.26667</td>

  <td align="right">0.41805282</td>
  <td align="right">0.20718125</td>
  <td align="right">1.37476593</td>
</tr>
<tr>
  <td align="right">1.36444</td>
  <td align="right">0.09518</td>
  <td align="right">+0.05632</td>
  <td align="right">+0.40707</td>
  <td align="right">-6.68275</td>
  <td align="right">-1.24612</td>

  <td align="right">-118.657</td>
  <td align="right">-3.06119</td>

  <td align="right">+0.63035</td>
  <td align="right">+0.59414</td>
  <td align="right">-0.59714</td>
  <td align="right">-0.85376</td>

  <td align="right">-0.94731</td>
  <td align="right">-1.43697</td>

  <td align="right">0.14374587</td>
  <td align="right">0.09152713</td>
  <td align="right">1.76472699</td>
</tr>
<tr>
  <td align="right">1.69351</td>
  <td align="right">0.11813</td>
  <td align="right">+0.15764</td>
  <td align="right">+0.38504</td>
  <td align="right">-6.27092</td>
  <td align="right">-1.02536</td>

  <td align="right">-39.7800</td>
  <td align="right">-2.66300</td>

  <td align="right">+0.73061</td>
  <td align="right">+0.44893</td>
  <td align="right">-1.35309</td>
  <td align="right">-0.87944</td>

  <td align="right">-1.85200</td>
  <td align="right">-1.95897</td>

  <td align="right">0.18178501</td>
  <td align="right">0.08464699</td>
  <td align="right">1.73356799</td>
</tr>
<tr>
  <td align="right">1.52303</td>
  <td align="right">0.05315</td>
  <td align="right">+0.03311</td>
  <td align="right">+0.29600</td>
  <td align="right">-9.85239</td>
  <td align="right">-0.84580</td>

  <td align="right">-297.565</td>
  <td align="right">-2.85743</td>

  <td align="right">+0.52221</td>
  <td align="right">+0.38805</td>
  <td align="right">-0.62474</td>
  <td align="right">-0.64518</td>

  <td align="right">-1.19634</td>
  <td align="right">-1.66262</td>

  <td align="right">0.08593434</td>
  <td align="right">0.04618515</td>
  <td align="right">1.86788051</td>
</tr>
<tr>
  <td align="right">1.78590</td>
  <td align="right">0.06233</td>
  <td align="right">+0.08952</td>
  <td align="right">+0.28558</td>
  <td align="right">-9.19424</td>
  <td align="right">-0.74657</td>

  <td align="right">-102.706</td>
  <td align="right">-2.61422</td>

  <td align="right">+0.57083</td>
  <td align="right">+0.31825</td>
  <td align="right">-1.4418</td>
  <td align="right">-0.66992</td>

  <td align="right">-2.52580</td>
  <td align="right">-2.10501</td>

  <td align="right">0.10258739</td>
  <td align="right">0.04358267</td>
  <td align="right">1.85382994</td>
</tr>
<tr>
  <td align="left" colspan="19">NOTE: &nbsp; All frequencies are given in the unit of <math>(\pi G \rho)^{1 / 2}</math>.</td>
</tr>
</table>


<span id="ReverseEngineered">Given the values</span> of <math>\Omega_2</math> and <math>\Omega_3</math> from this table, we have reverse-engineered this problem and determined what numerical values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> were used by {{ Chandrasekhar66_XXVIII }} for various models. What follows are the expressions that have been derived via this reverse-engineering effort.  The values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> that we have derived in this manner have been recorded in the last three columns of the table.

First &hellip;

<table align="center" border=0 cellpadding="3">

<tr>
  <td align="right">
<math>
\beta \biggl( \frac{\Omega_2^2}{\pi G \rho} \biggr) \biggl[ \frac{b^2 - c^2}{2c^2} \biggr]
\biggl[ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  
\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a^2(3b^2-4a^2 + c^2)\biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] + a^2 b^2 \biggl[ 2 - (A_2 + A_3) \biggr] - b^2c^2 A_3
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\beta}{4c^2} \biggl( \frac{\Omega_2^2}{\pi G \rho} \biggr) 
\underbrace{(b^2 - c^2)^2 \biggl[ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 \biggr]}_{\Gamma}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a^2(3b^2-4a^2 + c^2)\biggl[ A_2 b^2 - A_3 c^2 \biggr] + a^2 b^2 (b^2 - c^2)\biggl[ 2 - (A_2 + A_3) \biggr] - b^2c^2(b^2 - c^2) A_3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2a^2 b^2 (b^2 - c^2) 
+ a^2b^2(3b^2-4a^2 + c^2) A_2  
- a^2 b^2 (b^2 - c^2)A_2  
- a^2 c^2 (3b^2-4a^2 + c^2) A_3  
- a^2 b^2 (b^2 - c^2)A_3  
- b^2c^2(b^2 - c^2) A_3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2a^2 b^2 (b^2 - c^2) 
+ a^2b^2 [(3b^2-4a^2 + c^2)   
- (b^2 - c^2) ] A_2 
- [a^2 c^2 (3b^2-4a^2 + c^2)  
+ a^2 b^2 (b^2 - c^2) 
+ b^2c^2(b^2 - c^2) ]A_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2a^2 b^2 (b^2 - c^2) 
+ 2 a^2b^2 [ b^2- 2a^2 + c^2 ] A_2 
- [
(3 a^2 b^2 c^2 - 4a^4c^2 + a^2 c^4)  
+ (a^2 b^4 - a^2 b^2 c^2) 
+ (b^4 c^2 - b^2 c^4) 
]A_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\underbrace{2a^2 b^2 (b^2 - c^2)}_{\Lambda_{20}} 
+ \underbrace{2 a^2b^2 [ b^2- 2a^2 + c^2 ]}_{\Lambda_{22}} A_2 
- \underbrace{[ 2 a^2 b^2 c^2 - 4a^4c^2 + a^2 c^4  + a^2 b^4  + b^4 c^2 - b^2 c^4 ]}_{\Lambda_{23}}A_3 
\, .
</math>
  </td>
</tr>
</table>

Next &hellip;

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

<tr>
  <td align="right">
<math>
\gamma \biggl( \frac{\Omega_3^2}{\pi G \rho}\biggr) \biggl[ \frac{c^2 - b^2}{4b^2} \biggr]
\biggl[ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 
\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ a^2( b^2 + 3c^2 - 4a^2 ) B_{23} +c^2(a^2A_1 - b^2 A_2)  
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a^2( b^2 + 3c^2 - 4a^2 ) \biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] 
+ a^2c^2 \biggl[ 2 - (A_2 + A_3) \biggr] - b^2 c^2 A_2  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
-~ \frac{\gamma}{4b^2} \biggl( \frac{\Omega_3^2}{\pi G \rho}\biggr) 
\underbrace{(b^2 - c^2)^2\biggl[ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2 \biggr]}_\Gamma
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a^2( b^2 + 3c^2 - 4a^2 ) \biggl[ A_2 b^2 - A_3 c^2 \biggr] 
+ a^2c^2 (b^2 - c^2) \biggl[ 2 - (A_2 + A_3) \biggr] - b^2 c^2  (b^2 - c^2)A_2  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2a^2c^2 (b^2 - c^2)  
+ a^2 b^2 ( b^2 + 3c^2 - 4a^2 ) A_2  
- a^2c^2 (b^2 - c^2) A_2  
- b^2 c^2  (b^2 - c^2)A_2  
- a^2 c^2 ( b^2 + 3c^2 - 4a^2 ) A_3 
- a^2c^2 (b^2 - c^2) A_3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2a^2c^2 (b^2 - c^2)  
+ [ a^2 b^2 ( b^2 + 3c^2 - 4a^2 )   
- a^2c^2 (b^2 - c^2)   
- b^2 c^2  (b^2 - c^2) ]A_2  
- 2a^2 c^2 [ b^2 + c^2 - 2a^2  ] A_3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\underbrace{2a^2c^2 (b^2 - c^2)}_{\Lambda_{30}}  
+ \underbrace{[ 2a^2 b^2 c^2 - 4a^4 b^2 + a^2 b^4 + a^2 c^4 - b^4 c^2 + b^2 c^4 ]}_{\Lambda_{32}}A_2  
- \underbrace{2a^2 c^2 [ b^2 + c^2 - 2a^2  ]}_{\Lambda_{33}} A_3  \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>
\frac{\beta}{4c^2} \biggl( \frac{\Omega_2^2}{\pi G \rho} \biggr) \Gamma
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Lambda_{20} + \Lambda_{22} A_2 - \Lambda_{23} A_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ A_3 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\Lambda_{23} }\biggl[ 
\Lambda_{20} + \Lambda_{22} A_2 
- \frac{\beta}{4c^2} \biggl( \frac{\Omega_2^2}{\pi G \rho} \biggr) \Gamma
\biggr] \, ;
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>
-~ \frac{\gamma}{4b^2} \biggl( \frac{\Omega_3^2}{\pi G \rho}\biggr) \Gamma
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Lambda_{30} + \Lambda_{32} A_2 - \Lambda_{33} A_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\Lambda_{33}} \biggl[
\Lambda_{30} + \Lambda_{32} A_2 + \frac{\gamma}{4b^2} \biggl( \frac{\Omega_3^2}{\pi G \rho}\biggr) \Gamma
\biggr] \, .
</math>
  </td>
</tr>
</table>

Put together, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> 
\biggl[ \frac{\Lambda_{22}}{\Lambda_{23}} \biggr] A_2 +
\frac{1}{\Lambda_{23} }\biggl[ 
\Lambda_{20}  
- \frac{\beta}{4c^2} \biggl( \frac{\Omega_2^2}{\pi G \rho} \biggr) \Gamma
\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\Lambda_{32}}{\Lambda_{33}} \biggr]A_2
+
\frac{1}{\Lambda_{33}} \biggl[
\Lambda_{30}  + \frac{\gamma}{4b^2} \biggl( \frac{\Omega_3^2}{\pi G \rho}\biggr) \Gamma
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math> \Rightarrow ~~~
A_2 \biggl[ \frac{\Lambda_{22}}{\Lambda_{23}} - \frac{\Lambda_{32}}{\Lambda_{33}} \biggr] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\Lambda_{33}} \biggl[
\Lambda_{30}  + \frac{\gamma}{4b^2} \biggl( \frac{\Omega_3^2}{\pi G \rho}\biggr) \Gamma
\biggr] 
-
\frac{1}{\Lambda_{23} }\biggl[ 
\Lambda_{20}  
- \frac{\beta}{4c^2} \biggl( \frac{\Omega_2^2}{\pi G \rho} \biggr) \Gamma
\biggr]

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


<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center" rowspan="2"><math>\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>\frac{a_3}{a_1}</math></td>
  <td align="center" rowspan="2"><math>A_1</math></td>
  <td align="center" rowspan="2"><math>A_2</math></td>
  <td align="center" rowspan="2"><math>A_3</math></td>
  <td align="center" rowspan="3" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" rowspan="1" colspan="4">Direct</td>
  <td align="center" rowspan="3" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" rowspan="1" colspan="4">Adjoint</td>
</tr>
<tr>
  <td align="center"><math>\beta_+</math></td>
  <td align="center"><math>\gamma_+</math></td>
  <td align="center"><math>-\beta_+ (a^2+c^2)/c^2</math></td>
  <td align="center"><math>-\gamma_+ (a^2+b^2)/b^2</math></td>

  <td align="center"><math>\beta_-</math></td>
  <td align="center"><math>\gamma_-</math></td>
  <td align="center"><math>-\beta_-(a^2+c^2)/c^2</math></td>
  <td align="center"><math>-\gamma_- (a^2+b^2)/b^2</math></td>
</tr>
<tr>
  <td align="center">1.25000</td>
  <td align="center">0.50000</td>

  <td align="center">0.50823343</td>
  <td align="center">0.37944073</td>
  <td align="center">1.11232585</td>

  <td align="center">1.12066896</td>
  <td align="center">1.77691896</td>

  <td align="center" bgcolor="yellow">-5.6033448</td>
  <td align="center" bgcolor="yellow">-2.9141471</td>

  <td align="center">0.22308104</td>
  <td align="center">0.87933104</td>

  <td align="center" bgcolor="pink">-1.1154052</td>
  <td align="center" bgcolor="pink">-1.4421029</td>
</tr>
</table>

===Examination of Lagrangian Flow in One Specific Model===

This particular set of seven key parameters has been drawn from [[Appendix/References#EFE|[<font color="red">EFE</font>] ]] Chapter 7, Table XIII (p. 170). The tabular layout presented here, also appears in a [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model|related discussion labeled, ''Challenges Pt. 2'']].
<table width="80%" align="center" cellpadding="8" border="0">
<tr><td align="left"><math>~a = a_1 = 1</math></td></tr>
<tr><td align="left"><math>~b = a_2 = 1.25</math></td></tr>
<tr><td align="left"><math>~c = a_3 = 0.4703</math></td></tr>
<tr><td align="left"><math>~\Omega_2 = 0.3639</math></td></tr>
<tr><td align="left"><math>~\Omega_3 = 0.6633</math></td></tr>
<tr><td align="left"><math>~\zeta_2 = - 2.2794</math></td></tr>
<tr><td align="left"><math>~\zeta_3 = - 1.9637</math></td></tr>
</table>

As a consequence &#8212; see [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|an accompanying discussion]] (alternatively, [[ThreeDimensionalConfigurations/ChallengesPt6#Are_Orbits_Exact_Circles|ChallengesPt6]]) for details &#8212; the values of other parameters are &hellip;

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="4">&nbsp;</td>
  <td align="center" rowspan="8" colspan="1" bgcolor="lightgrey">&nbsp; </td>
  <td align="center" colspan="2">'''Example Values'''</td>
</tr>

<tr>
  <td align="right">
<math>~\tan\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\zeta_2 }{ \zeta_3 } \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]\frac{c^2}{b^2} = -0.344793</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~~ \theta =</math>
  </td>
  <td align="left">
<math>~- 19.0238^\circ</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Lambda
</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \cos\theta    
-
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\Lambda =</math>
  </td>
  <td align="left">
<math>~-1.332892 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\frac{y_0}{z_0}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Lambda} 
=
- \frac{b^2 \sin\theta}{(c^2\cos^2\theta + b^2\sin^2\theta)}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{y_0}{z_0} =</math>
  </td>
  <td align="left">
<math>~+ 1.400377</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\frac{x_\mathrm{max}}{ y_\mathrm{max} }   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \Lambda \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} \biggr\}^{1 / 2}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
&nbsp; &nbsp; <math>~\frac{x_\mathrm{max}}{y_\mathrm{max}} =</math>
  </td>
  <td align="left">
<math>~+ 1.025854</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{a(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2}}{bc}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
&nbsp; &nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>~
|\dot\varphi|
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
</math> &nbsp; <font color="green">Correct!</font>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>|\dot\varphi| =</math>
  </td>
  <td align="left">
<math>1.299300</math> &nbsp; <font color="green">Correct!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\dot\varphi
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{c^2(a^2 + b^2)}  \biggr] 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\dot\varphi =</math>
  </td>
  <td align="left">
<math>~-1.299300</math>
  </td>
</tr>
</table>

==EFE Rotating Cartesian Frame==
Concentric triaxial ellipsoids are defined by the expression,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>\biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2 \, ,</math></td>
</tr>
</table>
where <math>0 \le P \le 1</math> is a constant.  As viewed from the rotating reference frame, the velocity flow-field everywhere inside <math>(0 \le P < 1)</math>, and on the surface <math>(P = 1)</math> of the Type I Riemann ellipsoid is given by the expression &#8212; see, for example, an [[ThreeDimensionalConfigurations/ChallengesPt6#Riemann_Flow|accompanying discussion of the Riemann flow-field]],

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathbf{u}_\mathrm{EFE}</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>
\boldsymbol{\hat\imath} \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \biggr\}
+
\boldsymbol{\hat\jmath} \biggl\{ +\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \biggr\}
+
\mathbf{\hat{k}} \biggl\{ - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x  \biggr\}
\, .</math>
  </td>
</tr>
</table>
In an [[ThreeDimensionalConfigurations/ChallengesPt6#EFE_Rotating_Frame|accompanying discussion]], we have shown that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathbf{u}_\mathrm{EFE} \cdot \nabla P</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>0 \, ,</math></td>
</tr>
</table>
which means that, at every location inside and on the surface of the configuration, the velocity vector is orthogonal to a vector that is normal to the ellipsoidal surface at that location.

==Tilted Coordinate System==

<table border="1" align="center" width="60%" cellpadding="8">
<tr>
  <td align="center" colspan="8">'''Figure 1: &nbsp; Tilted Reference Frame'''</td>
</tr>
<tr>
<td align="left">

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

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\imath'} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\jmath}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath'}\cos\theta - \boldsymbol{\hat{k}'}\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath'}\sin\theta + \boldsymbol{\hat{k}'}\cos\theta \, .</math>
  </td>
</tr>
</table>
</td>
<td align="center">[[File:PrimedCoordinates3.png|250px|Primed Coordinates]]</td>
<td align="left">

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

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~x' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~y' \cos\theta - z' \sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z - z_0</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~y' \sin\theta + z'\cos\theta \, .</math>
  </td>
</tr>
</table>

</td></tr></table>

As we have detailed in our [[ThreeDimensionalConfigurations/ChallengesPt6#For_Arbitrary_Tip_Angles|accompanying discussion]], as viewed from this "tipped"  frame, the concentric ellipsoidal surfaces of a Type I Riemann ellipsoid are defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{y'\cos\theta - z'\sin\theta}{b}\biggr]^2 
+ \biggl[\frac{z_0 + z'\cos\theta + y'\sin\theta}{c}\biggr]^2
+\biggl(\frac{x'}{a}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>

and the <span id="CompactFlowField">velocity flow-field</span> is given by the expression,

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

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{ 
- \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[\boldsymbol{\hat\jmath'} \cos\theta  - \mathbf{\hat{k}'} \sin\theta \biggr] \biggl\{ 
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 x'
\biggr\}
+
\biggl[\boldsymbol{\hat\jmath'} \sin\theta  + \mathbf{\hat{k}'} \cos\theta \biggr] \biggl\{ 
- 
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x'
\biggr\} \, .
</math>
  </td>
</tr>
</table>
We also have explicitly demonstrated that, for any arbitrarily chosen value of the tilt angle, <math>\theta</math>,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathbf{u'}_\mathrm{EFE} \cdot \nabla P'</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>0 \, .</math></td>
</tr>
</table>

==Preferred Tilt==

As we discuss [[ThreeDimensionalConfigurations/ChallengesPt6#For_Specific_Tip_Angle|elsewhere]], if we specifically choose,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\beta \Omega_2}{\gamma \Omega_3} = - \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, .
</math>
  </td>
</tr>
</table>
the component of the flow-field in the <math>\mathbf{\hat{k}'}</math> direction vanishes; that is, in this specific case, as viewed from the tilted reference frame, all of the fluid motion is confined to the x'-y' plane.  Notice that this plane is not parallel to any of the three principal planes of the Type I Riemann ellipsoid.  <font color="red">I have not seen this fluid-flow behavior previously described in the published literature.  Maybe Norman Lebovitz will know.</font>

The three panels of Figure 2, and the text description that follows, have been drawn from a [[ThreeDimensionalConfigurations/ChallengesPt2#COLLADA-Based_Representation|separate discussion]].


<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center">Figure 2a</th>
  <th align="center">Figure 2b</th>
</tr>
<tr>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470B.cropped.png|500px|EFE Model b41c385]]
  </td>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470A.cropped.png|500px|EFE Model b41c385]]
  </td>
</tr>
<tr>
  <td align="center" colspan="2" bgcolor="lightgrey">
[[File:DataFileButton02.png|75px|file = Dropbox/3Dviewers/AutoRiemann/TypeI/Lagrange/TL15.lagrange.dae]] <font size="+2">&#x21b2;</font>
  </td>
</tr>
<tr>
  <th align="center" colspan="2">Figure 2c</th>
</tr>
<tr>
  <td align="center" bgcolor="white" colspan="2">
[[File:ProjectedOrbitsFlipped2.png|600px|EFE Model b41c385]]<br />
  <div align="center">[[File:DataFileButton02.png|75px|file = Dropbox/3Dviewers/RiemannModels/RiemannCalculations.xlsx --- worksheet = TypeI_1b]] <font size="+2">&#x21b2;</font></div>
  </td>
</tr>
</table>
</div>

As has been described in an [[ThreeDimensionalConfigurations/RiemannTypeI#Figure3|accompanying discussion of Riemann Type 1 ellipsoids]], we have used COLLADA to construct  an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 2 show what this ellipsoid looks like when viewed from two different perspectives.  (As a reminder &#8212; see the [[#explanation| explanation accompanying Figure 2 of that accompanying discussion]] &#8212; the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25&deg; to the equilibrium spin axis, which is shown in green.)  Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the [[#ExampleTrajectories|table that accompanies that discussion]].  From the perspective presented in Figure 2b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310.  When viewed from the perspective presented in Figure 2a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of &theta; = -19.02&deg; to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.

==Lagrangian Fluid Trajectories==

===Off-Center Ellipse===

The yellow dots in Figures 2a and 2b trace three different, nearly circular, closed curves.  These curves each show what results from the intersection of the surface of the triaxial ellipsoid and a plane that is tilted with respect to the x = x' axis by the specially chosen angle, <math>\theta</math>; the different curves result from different choices of the intersection point, <math>z_0</math>.  Several additional such curves are displayed in Figure 2c.  Each of these curves necessarily also identifies the trajectory that is followed by a fluid element that sits on the surface of the ellipsoid.  

We have determined that the <math>y'(x')</math> function that defines each closed curve is describable analytically by the expression,

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

<tr>
  <td align="right">
<math>
\biggl[ \frac{y' - y'_0}{y'_\mathrm{max}} \biggr]^2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \biggl( \frac{x'}{x'_\mathrm{max}} \biggr)^2 \, ,
</math>
  </td>
</tr>
</table>
where (see independent derivations with identical results from [[ThreeDimensionalConfigurations/ChallengesPt2#OffCenter|ChallengesPt2]] and [[ThreeDimensionalConfigurations/ChallengesPt6#Are_Orbits_Exact_Circles|ChallengesPt6]]),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> x'_\mathrm{max}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a\biggl[ 1 -\frac{z_0^2 \cos^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )} \biggr]^{1 / 2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a\biggl[ \frac{(c^2 - z_0^2) \cos^2\theta + b^2\sin^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )} \biggr]^{1 / 2} \, ,  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>y'_\mathrm{max} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
bc \biggl[ (c^2  \cos^2\theta   + b^2 \sin^2\theta )  - z_0^2 \cos^2\theta   \biggr]^{1 / 2} ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{-1}
 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
b c \biggl[ \frac{(c^2 - z_0^2) \cos^2\theta + b^2\sin^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )^2} \biggr]^{1 / 2} \, ,  
 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>y'_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{z_0 b^2\sin\theta }{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )} \, .
</math>
  </td>
</tr>

</table>

This is the equation that describes a closed ellipse with semi-axes, <math>(x'_\mathrm{max}, y'_\mathrm{max})</math>, that is offset from the z'-axis along the y'-axis by a distance, <math>y'_0</math>.  Notice that the degree of flattening,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{x'}{y'} \biggr]_\mathrm{max} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{bc} \, ,
</math>
  </td>
</tr>
</table>
is independent of <math>z_0</math>; that is to say, the degree of flattening of all of the elliptical trajectories is identical!  Notice, as well, that the y-offset, <math>y'_0</math>, is linearly proportional to <math>z_0</math>.

In a [[ThreeDimensionalConfigurations/ChallengesPt6#Plot_Off-Center,_Slightly_Flattened_Ellipse|separate discussion]], we have demonstrated that the [[#CompactFlowField|compact version of the ''tilted'' flow-field]] is everywhere orthogonal to the elliptical trajectory whose analytic definition is given by the off-set ellipse equation.

===Associated Lagrangian Velocities===
Let's presume that, as a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y' - y_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_0 - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\dot{y}' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
  </td>
</tr>
</table>

If this is the correct description of the Lagrangian motion in a <math>z' = 0</math> plane of motion, then the velocity components, <math>\dot{x}'</math> and <math>\dot{y}'</math>, must match the [[ThreeDimensionalConfigurations/ChallengesPt6#SpecificTipAngle|respective components of the Riemann flow-field]], namely,

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

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}\biggr|_{z'=0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
+
\boldsymbol{\hat\imath'} \biggl\{ 
\cancelto{0}{z' (c^2 - b^2 )\tan\theta}  
- y' [c^2  + b^2 \tan^2\theta ]
\biggr\}\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
+
\boldsymbol{\hat\jmath'} \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3~x' }{\cos\theta} \, .
</math>
  </td>
</tr>
</table>
First, let's compare the <math>\boldsymbol{\hat\jmath'}</math> components.

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

<tr>
  <td align="right">
<math>x' \biggl[ \frac{y'}{x'}\biggr]_\mathrm{max} \dot\varphi</math>
  </td>
  <td align="center">
<math>\leftrightarrow</math>
  </td>
  <td align="left">
<math>
x' \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3}{\cos\theta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3}{\cos\theta}\biggl[ \frac{x'}{y'}\biggr]_\mathrm{max} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3}{\cos\theta}
\biggl[\frac{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{bc}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{c^2(a^2 + b^2)}  \biggr] 
\, .
</math>
  </td>
</tr>
</table>

Now let's compare the <math>\boldsymbol{\hat\imath'}</math> components.

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

<tr>
  <td align="right">
<math>(y_0 - y') \biggl[ \frac{x'}{y'}\biggr]_\mathrm{max} \dot\varphi</math>
  </td>
  <td align="center">
<math>\leftrightarrow</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' [c^2  + b^2 \tan^2\theta ]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ (y_0 - y') \dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' [c^2  + b^2 \tan^2\theta ]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\} \biggl[ \frac{y'}{x'}\biggr]_\mathrm{max} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\biggl[\frac{bc}{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}  \biggr] \, .
</math>
  </td>
</tr>
</table>
Inserting the just-derived expression for <math>\dot\varphi</math> into this last expression gives,

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

<tr>
  <td align="right">
<math>(y_0 - y') </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\biggl[\frac{bc}{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}  \biggr]
\biggl[ \frac{a^2 + b^2}{b^2}\biggr] \frac{\cos\theta}{\zeta_3}
\biggl[\frac{bc}{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\frac{\cos\theta}{\zeta_3}
\biggl[\frac{c^2 (a^2 + b^2)}{ a^2( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}  
\frac{\cos\theta}{\zeta_3}
\biggl[\frac{c^2 (a^2 + b^2)}{ a^2( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
+
\biggl\{ 
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\frac{\cos\theta}{\zeta_3}
\biggl[\frac{c^2 (a^2 + b^2)}{ a^2( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr] \frac{\zeta_2 }{\zeta_3}
\biggr\}  
\biggl[\frac{b^2 z_0 \cos\theta}{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
-
y' \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, .
</math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>(y_0 - y') </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{- b^2 z_0 \sin\theta}{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
-
y' \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ y_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{- b^2 z_0 \sin\theta}{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
\, .
</math>
  </td>
</tr>
</table>

<font color="red">SUCCESS !!!</font>

=See Also=

* [[3Dconfigurations/DescriptionOfRiemannTypeI|Description of Riemann Type I Ellipsoids]] (best)
* [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI|Description of Riemann Type I Ellipsoids]] (older introduction)
* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|Riemann Type 1 Ellipsoids]] (oldest introduction)
* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Construction Challenges (Pt. 1)]]
* [[ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[ThreeDimensionalConfigurations/ChallengesPt3|Construction Challenges (Pt. 3)]]
* [[ThreeDimensionalConfigurations/ChallengesPt4|Construction Challenges (Pt. 4)]]
* [[ThreeDimensionalConfigurations/ChallengesPt5|Construction Challenges (Pt. 5)]]
* [[ThreeDimensionalConfigurations/ChallengesPt6|Construction Challenges (Pt. 6)]]

* Related discussions of models viewed from a rotating reference frame:
** [[PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]
** <font color="red"><b>NOTE to Eric Hirschmann &amp; David Neilsen...  </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoidsCompressible|Compressible Riemann Ellipsoids]].

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