Editing ThreeDimensionalConfigurations/RiemannTypeI (section)

====Methodical Derivation of Orbital Parameters====
Let the unprimed coordinates (x, y, z) represent the body frame of the ellipsoid, and the primed coordinates (x', y', z') represent a coordinate system in which the z'-axis is "tipped" &#8212; about the x' = x axis &#8212; away from the z-axis by an angle, &theta;.  We assume that the motion of each Lagrangian fluid element will be restricted to an x'-y' equatorial plane &#8212; that is, we assume that <math>~\dot{z}' \equiv dz'/dt = 0</math> &#8212; but in general its velocity vector in the unprimed "body" frame will have the three nonzero components as specified in EFE and above.  Here are some relevant transformations between these two coordinate systems.
<table border="1" align="center" cellpadding="10">
<tr>
<td align="center">
<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 \, ,</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>~(z-z_0)\cos\theta - y\sin\theta \, .</math>
  </td>
</tr>
</table>

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

<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' \, ,</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>~z'\cos\theta + y'\sin\theta \, .</math>
  </td>
</tr>
</table>

</td>
</tr>

<tr>
<td align="center">
<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>~\dot{x} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{y} \cos\theta + \dot{z}\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cancelto{0}{\dot{z}'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{z} \cos\theta - \dot{y}\sin\theta \, .</math>
  </td>
</tr>
</table>

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

<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>~\dot{x}' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{y}' \cos\theta - \cancelto{0}{\dot{z}'}\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cancelto{0}{\dot{z}'}\cos\theta + \dot{y}'\sin\theta \, .</math>
  </td>
</tr>
</table>

</td>
</tr>

</table>


<b><font color="red">NOTE:</font></b>  The center of each elliptical orbit is (x', y', z') = (0, y<sub>0</sub>, 0).  In "body" coordinates, then, (x, y, z) = (0, y<sub>0</sub> cos&theta;, z<sub>0</sub> + y<sub>0</sub> sin&theta;).

Focusing on the bottom-right quadrant of this equation-table, we note first that <math>~(\dot{x}, \dot{y}, \dot{z}) = (u_1, u_2, u_3)</math>, as [[#EFEvelocities|provided above from EFE]].  Copying from the upper-right quadrant of this equation-table, let's rewrite these three velocity components in terms of the "tipped" plane coordinates, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\dot{x} = 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 \biggl[ y'\cos\theta - z'\sin\theta\biggr] 
+ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[ z_0 + z'\cos\theta + y'\sin\theta \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y} = 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>~\dot{z} = 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>


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>

The first of these equation pairs can be plugged directly into the expressions for u<sub>1</sub>, u<sub>2</sub>, and u<sub>3</sub> &#8212; further fleshing out the LHS of the equations in the bottom-right quadrant of the above equation-table &#8212; while the second pair can be used to re-express the RHS of the equations in the bottom-right quadrant of this equation table.  We obtain,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \biggl\{  \biggl[ y_0 + y_\mathrm{max} \sin(\varphi t) \biggr] \cos\theta - \cancelto{0}{z'}\sin\theta \biggr\} 
+ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl\{ z_0 + \cancelto{0}{z'} \cos\theta + \biggl[ y_0 + y_\mathrm{max} \sin(\varphi t) \biggr]\sin\theta \biggr\} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x_\mathrm{xmax} ~\varphi \cdot \sin(\varphi t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x_\mathrm{max}\cos(\varphi t) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}~\varphi \cdot \cos(\varphi t) \cdot \cos\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x_\mathrm{max}\cos(\varphi t) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}~\varphi \cdot \cos(\varphi t) \cdot \sin\theta \, .</math>
  </td>
</tr>
</table>

Combining the second and third of these conditions, we find first that,

<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{\zeta_2 }{ \zeta_3 } \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]\frac{c^2}{b^2} \, ,</math>
  </td>
</tr>
</table>

which gives the "tipping" angle in terms of known parameter values; and second that &hellip;  
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\varphi \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]  
</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} 
\, ,</math>
  </td>
</tr>
</table>
which gives the product of the two unknown quantities, <math>~\varphi</math> and the ratio <math>~y_\mathrm{max}/x_\mathrm{max}</math>, in terms of known parameter values.  And from the first condition, we furthermore find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-x_\mathrm{max} ~\varphi \cdot \sin(\varphi t) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \biggl[ y_0 \cos\theta \biggr]   
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \biggl[ y_\mathrm{max} \sin(\varphi t) \biggr] \cos\theta 
+ 
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[ z_0 + y_0 \sin\theta \biggr]
+ 
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[ y_\mathrm{max} \sin(\varphi t) \biggr]\sin\theta  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~ x_\mathrm{max} ~\varphi \cdot \sin(\varphi t) 
+ \biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta  
- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \cos\theta \biggr\} \biggl[ y_\mathrm{max} \sin(\varphi t) \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{
\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \biggl[ y_0 \cos\theta \biggr]   
-
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z_0 
-
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl[ y_0 \sin\theta \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
The LHS and the RHS must separately sum to zero, which means &hellip;
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\biggl\{ \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  \biggr\}y_0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z_0 \, ,
</math>
  </td>
</tr>
</table>

which gives the ratio, <math>~y_0/z_0</math> in terms of known parameter values; and,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\biggl[ \frac{x_\mathrm{max}}{ y_\mathrm{max} } \biggr] ~\varphi  
</math>
  </td>
  <td align="center">
<math>~=</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>
</tr>
</table>
which in combination with the just-derived similar product relation can give expressions for both terms, namely,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\biggl[ \frac{x_\mathrm{max}}{ y_\mathrm{max} } \biggr]^2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \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  \biggr\}\biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} 
\, ,
</math>
  </td>
</tr>
</table>
and,

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