Editing ThreeDimensionalConfigurations/ChallengesPt2 (section)

===Velocities===

====Tipped Orbit Velocities====
From the generic expressions for [[#Motivation|(primed) velocities associated with an off-center elliptical orbit]], we expect,

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

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

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

====Body Frame Velocities====

From the already-referenced table provided in [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|our accompanying discussion]], we can transform this pair of expressions for the velocity components in the "tipped orbit" frame &#8212; remember that the third component, <math>~\dot{z}' = 0</math> &#8212; into the (three-component) velocities of the body frame using the expressions,
<table border="1" width="75%" cellpadding="8" align="center">
<tr>
    <td align="left" width="50%">
<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
\, ,</math>
  </td>
</tr>

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

    </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-z_0)\sin\theta
\, ,</math>
  </td>
</tr>

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

    </td>
</tr>
</table>
That is to say,

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

<tr>
  <td align="right">
<math>~\frac{\dot{x}}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (y' - y_0) \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] 
=
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \biggl\{ y_0 - [y\cos\theta + (z-z_0)\sin\theta]  \biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\dot{y}}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \cos\theta
=
x \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \cos\theta \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\dot{z}}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \sin\theta
=
x \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \sin\theta \, ,
</math>
  </td>
</tr>
</table>
where,
<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>~
\frac{a^2}{b^2 c^2}\biggl[ c^2 \cos^2\theta + b^2\sin^2\theta \biggr] \, .
</math>
  </td>
</tr>
</table>
Notice that the all-important tipping angle, <math>~\theta</math>, is related to these body-frame velocity components via the simple relation,
<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{\dot{z}}{\dot{y}} \biggr) \, .</math>
  </td>
</tr>
</table>

====Vorticity Determination====
Given that the ratio, <math>~(x_\mathrm{max}/y_\mathrm{max})</math>, does not depend on <math>~z'</math>, and that, after mapping <math>~z_0 \rightarrow (z_0 + z'\cos\theta)</math>, we find,

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

<tr>
  <td align="right">
<math>~\frac{\partial y_0}{\partial z'} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial z'}
\biggl[
- \frac{(z_0 + z'\cos\theta) b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}
\biggr]
=
- \frac{ b^2 \sin\theta\cos\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}
=
- b^2 \sin\theta\cos\theta \biggl(\frac{a^2}{b^2c^2}\biggr) \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \, ,
</math>
  </td>
</tr>
</table>
the [[#VorticitySetup|above vorticity expression]] becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'} \equiv \boldsymbol{\nabla \times}\bold{v'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\boldsymbol{\hat\imath'} (x' \dot\varphi ) \cancelto{0}{\frac{\partial }{\partial z'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} }\biggr]}
+ \boldsymbol{\hat\jmath'} \biggl\{
\dot\varphi (y_0 - y') \cancelto{0}{\frac{\partial }{\partial z'}\biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr]}
+ \dot\varphi \biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr] \frac{\partial y_0}{\partial z'}
\biggr\}
+ \bold{\hat{k}'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} } - \frac{x_\mathrm{max} }{y_\mathrm{max} } \biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \boldsymbol{\hat\jmath'} 
\biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr] b^2 \sin\theta\cos\theta \biggl(\frac{a^2}{b^2c^2}\biggr) \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2\dot\varphi 
+ \bold{\hat{k}'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} } - \frac{x_\mathrm{max} }{y_\mathrm{max} } \biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[  \frac{y_\mathrm{max}}{x_\mathrm{max} } \biggr] \dot\varphi \biggl\{
- \boldsymbol{\hat\jmath'} 
~\sin\theta\cos\theta \biggl(\frac{a^2}{c^2}\biggr)  
+ \bold{\hat{k}'} \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
\biggr\} \, .
</math>
  </td>
</tr>
</table>

Referring back to our [[#Tipped_Orbital_Plane|aboved-defined tipped plane]], we see that the unprimed Cartesian unit vectors are related to the primed unit vectors via the relations &hellip;
<table border="1" align="center" cellpadding="10" width="60%"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

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

</td></tr></table>
Hence, from the perspective of the body frame, the expression for the vorticity becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[  \frac{y_\mathrm{max}}{x_\mathrm{max} } \biggr] \dot\varphi \biggl\{
- \biggl[ \boldsymbol{\hat{\jmath}}\cos\theta  + \boldsymbol{\hat{k}}\sin\theta  \biggr]
~\sin\theta\cos\theta \biggl(\frac{a^2}{c^2}\biggr)  
+ \biggl[ -\boldsymbol{\hat{\jmath}}\sin\theta  + \boldsymbol{\hat{k}}\cos\theta \biggr] \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\cos\theta}\biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] \biggl\{
- \boldsymbol{\hat{\jmath}} ~\sin\theta\cos^2\theta \biggl(\frac{a^2}{c^2}\biggr)  
- \boldsymbol{\hat{k}}~\sin^2\theta\cos\theta \biggl(\frac{a^2}{c^2}\biggr)  
-\boldsymbol{\hat{\jmath}}\sin\theta  \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
+ 
\boldsymbol{\hat{k}}\cos\theta \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] \biggl\{
- \boldsymbol{\hat{\jmath}} \biggl[  1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} }   
~+~\cos^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr]\tan\theta 
+ 
\boldsymbol{\hat{k}} \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} }  
~-~ \sin^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Now, we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 - \frac{x^2_\mathrm{max}}{y^2_\mathrm{max}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - \frac{a^2}{b^2 c^2}\biggl( c^2 \cos^2\theta + b^2\sin^2\theta \biggr)
=
1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] \biggl\{
- \boldsymbol{\hat{\jmath}} \biggl[  1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta   
~+~\cos^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr]\tan\theta 
+ 
\boldsymbol{\hat{k}} \biggl[ 1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta  
~-~ \sin^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~\boldsymbol{\hat{\jmath}} \biggl[  1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta   
~+~\cos^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr] \biggl[ \frac{c^2 \zeta_2}{a^2 + c^2} \biggr] 
+ 
\boldsymbol{\hat{k}} \biggl[ 1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta  
~-~ \sin^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr] \biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~\boldsymbol{\hat{\jmath}} \biggl\{  1 - \biggl( \frac{a^2}{b^2 c^2} \biggr) \biggl[ c^2 \cos^2\theta + b^2\sin^2\theta   
~-~b^2 \cos^2\theta  \biggr] \biggr\} 
\biggl[ \frac{c^2 \zeta_2}{a^2 + c^2} \biggr] 
+ 
\boldsymbol{\hat{k}} \biggl\{ 1 - \biggl( \frac{a^2}{b^2 c^2} \biggr) \biggl[ c^2 \cos^2\theta + b^2\sin^2\theta  
~+~ b^2 \sin^2\theta  \biggr] \biggr\}
\biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+~\boldsymbol{\hat{\jmath}} \biggl\{  c^2 - \biggl( \frac{a^2}{b^2} \biggr) \biggl[b^2 +  c^2 \cos^2\theta  ~-~2b^2 \cos^2\theta  \biggr] \biggr\} 
\biggl[ \frac{\zeta_2}{a^2 + c^2} \biggr] 
+ 
\boldsymbol{\hat{k}} \biggl\{ b^2 - \biggl( \frac{a^2}{c^2} \biggr) \biggl[c^2 - c^2 \sin^2\theta + 2b^2\sin^2\theta    \biggr] \biggr\}
\biggl[ \frac{\zeta_3}{a^2 + b^2} \biggr] 
</math>
  </td>
</tr>
</table>