Editing Appendix/Ramblings/T3CharacteristicVector (section)

==Implications==
Backing up to the expression that began our examination of the Binney and Tremaine formalism, we also can write,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right" colspan="2">
<math>
\frac{d\ln(h_2^2 \dot{\lambda}_2)}{dt}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="3">
<math>
\frac{\lambda_2}{\dot{\lambda}_2} \biggl[ \frac{\dot{\lambda}_1}{\lambda_1}  + \frac{\dot{\lambda}_2}{\lambda_2} \biggr]\frac{d\ln h_2}{dt} 
</math>
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</tr>

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<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
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<math>
\frac{d\ln(h_2^2 \dot{\lambda}_2)}{dt} \biggl(\frac{d\ln\lambda_2}{dt}\biggr)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="3">
<math>
\biggl[ \frac{d\ln(\lambda_1 \lambda_2)}{dt} \biggr]\frac{d\ln h_2}{dt} 
</math>
  </td>
</tr>
</table>


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<font color="red"><b>NOTE:</b>The following few ''boxed-in'' sentences/expressions are incorrect.</font> They originally appeared in this discussion due to confusion that arose in conjunction with a sign error in the expression for <math>d\ln C_2/dt</math> (see the top of this page).  On 14 July 2010, following a lengthy discussion between Joel and Jay of the statements colored in green, Jay spotted the sign error.  (See his [[Jaycall/KillingVectorApproach#Related_Talk_Session|14 July 2010 talk-page comment]].)  The proper derivation/conclusion resulting from the corrected sign error follows these ''boxed-in'' sentences/expressions.
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Comparing this with the ''brute force'' derivation of the condition derived above for the characteristic vector, <math>C_2</math>, we see that the two expressions are the same if we set,
<div align="center">
<math>
C_2 = h_2^2 \dot{\lambda}_2 .
</math>
</div>

<b><font color="darkgreen">
This seems to imply that we have discovered a conserved quantity, namely, <math>(h_2^2 \dot{\lambda}_2)^2</math>.  On the other hand, I might just be using a circular argument; I might only be saying that "the equation of motion is the equation of motion!"
</font></b>

Temp note (from Jay):  Joel, I don't quite understand this.  Next time we get together, can you explain this page to me?

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Comparing this last differential equation with the ''brute force'' derivation of the condition derived above for the characteristic vector, <math>C_2</math>, we see that the two expressions are the same if we set,
<div align="center">
<math>
C_2 = ( h_2^2 \dot{\lambda}_2 )^{-1} .
</math>
</div>

<b><font color="darkgreen">
At first sight, this seems to imply that we have discovered a conserved quantity.  But, alas, the result is a trivial one:  The resulting conserved quantity is, <math>C_2(h_2^2 \dot{\lambda}_2) = 1</math>.  
</font></b>