Editing Appendix/Mathematics/EulerAngles (section)

=Riemann Type I Ellipsoids=

==Tipped Coordinate System==
In an accompanying discussion of Riemann Type I ellipsoids, we have presented a [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|methodical derivation of orbital parameters]] in a "tipped" reference frame.  In the context of our present discussion of Euler angles, it seems clear that this separate coordinate-frame transformation involved only one nonzero Euler angle; specifically, <math>(\phi, \theta, \psi) = (0, \theta, 0)</math>.  In terms of Euler angles, then, the transformation between the ellipsoid's unprimed frame and the primed "tipped" frame is given by the [[#FormMatrix|above expressions]] &hellip;

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\begin{bmatrix}x' \\ y' \\ z' \end{bmatrix}_\mathrm{tipped}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\hat{R} \cdot
\begin{bmatrix}x \\ y \\ (z - z_0) \end{bmatrix} \, ,
</math>
  </td>
</tr>
</table>

and,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\begin{bmatrix}x \\ y \\ (z - z_0) \end{bmatrix}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\hat{R}^{-1} \cdot
\begin{bmatrix}x' \\ y' \\ z' \end{bmatrix}_\mathrm{tipped} \, ,
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>\hat{R}(\phi, \theta, \psi) = \hat{R}_3(\psi) \cdot \hat{R}_1(\theta) \cdot \hat{R}_3(\phi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
1 & 0 & 0  \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}} \cdot
{\begin{bmatrix}
1 & 0 & 0  \\
0 & \cos\theta & \sin\theta \\
0 & -\sin\theta & \cos\theta 
\end{bmatrix}} \cdot
{\begin{bmatrix}
1 & 0 & 0  \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
1 & 0 & 0  \\
0 & \cos\theta & \sin\theta \\
0 & -\sin\theta & \cos\theta 
\end{bmatrix}} \, ,
</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right">
<math>\hat{R}^{-1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
1 & 0 & 0  \\
0 & \cos\theta & -\sin\theta \\
0 & \sin\theta & \cos\theta 
\end{bmatrix}} \, .
</math>
  </td>
</tr>
</table>
That is to say,
<table border="1" align="center" cellpadding="8"><tr>
<td align="center">

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>x'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>x</math></td>
</tr>
<tr>
  <td align="right"><math>y'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>y \cos\theta + (z - z_0)\sin\theta</math></td>
</tr>
<tr>
  <td align="right"><math>z'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- y \sin\theta + (z - z_0)\cos\theta</math></td>
</tr>
</table>
</td>

<td align="center">
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>x'</math></td>
</tr>
<tr>
  <td align="right"><math>y</math></td>
  <td align="center"><math>=</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>=</math></td>
  <td align="left"><math>y' \sin\theta + z'\cos\theta</math></td>
</tr>
</table>
</td>
</tr>
</table>
which is identical to the coordinate transformations presented in the [[#Try_Again|top two panels of the table in our accompanying ''methodical derivation of orbital parameters]].  

Similarly, the three components of the velocity field as viewed from the "tipped" plane can be obtained from the velocity components in the untipped plane <math>(u_1, u_2, u_3)</math> via the expressions,
<table border="1" align="center" cellpadding="8"><tr>
<td align="center">

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>u_1^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>u_1</math></td>
</tr>
<tr>
  <td align="right"><math>u_2^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>u_2 \cos\theta + u_3\sin\theta</math></td>
</tr>
<tr>
  <td align="right"><math>u_3^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- u_2 \sin\theta + u_3\cos\theta</math></td>
</tr>
</table>
</td>
</tr>
</table>

==Velocity in Tipped Plane==
Now, according to EFE &#8212; see eq. 154 of Chapter 7, &sect;51 (p. 156)  &#8212;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~u_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~u_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x</math>
  </td>
</tr>
</table>

Hence, in the "tipped" plane we find,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>u_1^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y 
</math>
  </td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[z_0 + y'\sin\theta + z'\cos\theta\biggr]
+ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \biggl[ z' \sin\theta - y'\cos\theta \biggr] 
\, ,
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>u_2^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\cos\theta\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x  
- \sin\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x</math></td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl\{\cos\theta\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3   
- \sin\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2  \biggr\}x'
\, ,
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>u_3^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- \sin\theta\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x  -
\cos\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x</math></td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\biggl\{ - \sin\theta\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3   
- \cos\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \biggr\} x'
\, .
</math>
  </td>
</tr>
</table>
Ideally, we would like to choose the orientation of our "tipped" plane such that there are no vertical motions as viewed from that plane, that is, such that <math>u_3^' = 0</math>.  This means that,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\sin\theta\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
   - \cos\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>\Rightarrow ~~~ \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}   \, .
</math>
  </td>
</tr>
</table>
Given, as well, that we are only analyzing motion in the <math>z' = 0</math> plane, we have for the other two velocity components,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>u_2^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\cos\theta \biggl\{\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3   
- \tan\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2  \biggr\}x'
</math>
  </td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\cos\theta \biggl\{\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3   
+ \frac{\zeta_2}{\zeta_3}\biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]\frac{c^2}{b^2}
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2  \biggr\}x'
</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]\zeta_3 \cos\theta \biggl\{1   
+ \frac{\zeta_2^2}{\zeta_3^2}\biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]^2\frac{c^4}{b^4}
\biggr\}x'
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>u_1^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\cos\theta \biggl\{
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[\frac{z_0}{\cos\theta} + y'\tan\theta + \cancelto{0}{z'}\biggr]
+ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \biggl[ \cancelto{0}{z'} \tan\theta - y'\biggr] 
\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] z_0 \zeta_2 
- \zeta_3 \cos\theta \biggl\{
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] 
\frac{\zeta_2^2}{\zeta_3^2}\biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]\frac{c^2}{b^2}
+ \biggl[ \frac{a^2}{a^2 + b^2} \biggr]  
\biggr\}y'
</math>
  </td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] z_0 \zeta_2 
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr]\zeta_3 \cos\theta \biggl\{
\frac{\zeta_2^2}{\zeta_3^2}\biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]^2\frac{c^2}{b^2}
+ 1  
\biggr\}y'
\, .
</math>
  </td>
</tr>
</table>

==Rewrite Velocity-Component Expressions==
After defining the term,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Lambda^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{\zeta_2^2}{\zeta_3^2}\biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]^2\frac{c^2}{b^2} \, ,</math>
  </td>
</tr>
</table>
the expressions for the two "tipped" velocity components may be rewritten as,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] u_2^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl\{b^2    
+ \Lambda^2 \cdot c^2
\biggr\}x' \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] u_1^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] 
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] z_0 \zeta_2 
- \biggl\{ a^2 + \Lambda^2  \cdot a^2 
\biggr\}y'
\, .
</math>
  </td>
</tr>
</table>
Or,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\biggl\{\frac{1+\Lambda^2}{b^2    + \Lambda^2 \cdot c^2 }\biggr\}\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] u_2^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
( 1 + \Lambda^2 )x' 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\biggl[ \frac{a^2 + b^2}{a^2 \zeta_3 \cos\theta}\biggr] u_1^'</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] 
\biggl[ \frac{z_0 \zeta_2}{a^2 + c^2} \biggr]  
- ( 1 + \Lambda^2 )y'
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
( 1 + \Lambda^2 )(y_0 - y')
\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>y_0</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{1}{1+\Lambda^2}\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] 
\biggl[ \frac{z_0 \zeta_2}{a^2 + c^2} \biggr]  
\, .
</math>
  </td>
</tr>
</table>

==Lagrangian Particle Motion==

Now, if we assume that each Lagrangian particle executes a closed elliptical orbit ''in the plane'' of the tipped coordinate system (''i.e.,'' <math>~z' = \dot{z}' = 0</math> ), but whose orbit-center may be shifted by an amount, <math>~y_0</math>, away from the z'-axis, we expect &hellip;
<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(\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(\varphi t) \, ,</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- x_\mathrm{max}~ \varphi \cdot \sin(\varphi t)</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}~\varphi \cdot \cos(\varphi t) \, .</math>
  </td>
</tr>
</table>
That is to say, we expect that the two velocity components will exhibit the form, respectively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>u_1^'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \varphi \cdot (y_0 - y') \, ,
</math>
  </td>

<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>u_2^'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \varphi \cdot x' \, .
</math>
  </td>

</tr>
</table>
This pair of expressions will only match the underlying Riemann Type I flow if,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\biggl\{\frac{1+\Lambda^2}{b^2    + \Lambda^2 \cdot c^2 }\biggr\}\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{x_\mathrm{max}}{y_\mathrm{max}} 
\, ,
</math>
&nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right"><math>\biggl[ \frac{a^2 + b^2}{a^2 \zeta_3 \cos\theta}\biggr] </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{y_\mathrm{max}}{x_\mathrm{max}} \, .
</math>
  </td>
</tr>
</table>
That is,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\biggl\{\frac{1+\Lambda^2}{b^2    + \Lambda^2 \cdot c^2 }\biggr\}\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr] </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{a^2 \zeta_3 \cos\theta}{a^2 + b^2}\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{1}{a^2}\biggl\{\frac{1+\Lambda^2}{b^2    + \Lambda^2 \cdot c^2 }\biggr\}\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr]^2 </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
1  \, .
</math>
  </td>
</tr>
</table>
Let's try the numerical values from the [[ThreeDimensionalConfigurations/RiemannTypeI#Example_b1.25c0.470|example Type I ellipsoid used in an accompanying discussion]], namely, <math>b/a = 1.25, c/a = 0.4703, \zeta_2 = -2.2794, \zeta_3 = -1.9637</math>.

Hence,

<table border="0" align="center" cellpadding="8">

<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}   
= - 0.3448 ~~~ \Rightarrow ~~~ \theta = -0.3320 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \cos\theta</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
  - 0.9454 
\, .
</math>
  </td>
</tr>
</table>
And,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Lambda^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{\zeta_2^2}{\zeta_3^2}\biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]^2\frac{c^2}{b^2} 
= \tan^2\theta \cdot \frac{b^2}{c^2}
= 0.8399
\, .</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{1}{a^2}\biggl\{\frac{1+\Lambda^2}{b^2    + \Lambda^2 \cdot c^2 }\biggr\}\biggl[ \frac{a^2 + b^2}{\zeta_3 \cos\theta}\biggr]^2 </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{ 1.0524 \biggr\} \biggl[ 1.3803 \biggr]^2
=
2.005
\, .
</math>
  </td>
</tr>
</table>