Editing Appendix/Ramblings/T3CharacteristicVector

=Characteristic Vector for T3 Coordinates=
Let's apply Jay's [[Jaycall/KillingVectorApproach|Characteristic Vector approach]] to Joel's [[Appendix/Ramblings/T3Integrals|T3 Coordinate System]].

==Brute Force Manipulations==

Starting from '''[[Jaycall/KillingVectorApproach#CV.02|Equation CV.02]]''', and plugging in expressions for various [[Appendix/Ramblings/T3Integrals#Logarithmic_Derivatives_of_Scale_Factors|logarithmic derivatives of the T3 scale factors]], we obtain [<font color="red">Note: Sign error from equation CV.02 fixed here on 15 July 2010</font>],

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
<math>
- \frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2 \frac{\partial \ln h_1}{\partial\ln\lambda_2}  + \frac{\partial \ln h_2}{\partial \ln\lambda_2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2  \biggl( \frac{q h_1 h_2 \lambda_2}{\lambda_1 }  \biggr)^2  - ( qh_1^2 )^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ (h_1 \dot{\lambda}_1)^2  ( q h_1 h_2 \lambda_2 )^2  - (h_2 \dot{\lambda}_2)^{2} ( qh_1^2 \lambda_1  )^2  \biggr](h_2 \lambda_1 \dot{\lambda}_2)^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{\dot{\lambda}_1}{\lambda_1}\biggr)^2  - \biggl( \frac{\dot{\lambda}_2}{\lambda_2} \biggr)^2  \biggr]( q h_1^2 h_2 \lambda_1 \lambda_2 )^2 (h_2 \lambda_1 \dot{\lambda}_2)^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2}  \biggr] \biggl[ \frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2}  \biggr] \biggl( \frac{ q h_1^2 \lambda_2}{\dot{\lambda}_2} \biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;
  </td>
  <td align="right">
<math>
- \frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2}  \biggr] \biggl[ \frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2}  \biggr] ( q h_1^2 )^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2}  \biggr] \frac{d\ln h_2}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{d\ln(\lambda_1 \lambda_2)}{dt} \biggr] \frac{d\ln h_2}{dt}
</math>
  </td>
</tr>

<!--
<tr>
  <td align="center">
or
  </td>
  <td align="right">
<math>
\frac{\dot{C}_2}{C_2} 
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2}  \biggr] \frac{d\ln h_2}{dt} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\dot{\lambda}_1}{\lambda_1}\biggr] \frac{d\ln h_2}{dt} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1} + \frac{d\ln h_2}{dt}
</math>
  </td>
</tr>
-->

</table>

==Two Views of Equation of Motion==

===Christoffel Symbol Formalism===

The second component of the equation of motion can be obtained by setting <math>i = 2</math> and <math>C_i = 1</math> in '''[[User:Jaycall/KillingVectorApproach#CV.01|Equation CV.01]]''', specifically,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{d(h_2^2 \dot{\lambda}_2)}{dt}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
{h_k}^2 \Gamma^k_{2j} \dot{\lambda}_j \dot{\lambda}_k
</math>
  </td>
  <td align="left">
<math>
= {h_1}^2 \dot{\lambda}_1 \biggr[ \Gamma^1_{21} \dot{\lambda}_1 + \Gamma^1_{22} \dot{\lambda}_2 \biggl] + 
{h_2}^2 \dot{\lambda}_2 \biggr[ \Gamma^2_{21} \dot{\lambda}_1 + \Gamma^2_{22} \dot{\lambda}_2 \biggl] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="2">
<math>
{h_1}^2 \dot{\lambda}_1 \biggr[ \biggl( \frac{1}{h_1} \frac{\partial h_1}{\partial\lambda_2} \biggr) \dot{\lambda}_1 - \biggl( \frac{h_2}{h_1^2} \frac{\partial h_2}{\partial\lambda_1} \biggr) \dot{\lambda}_2 \biggl] + 
{h_2}^2 \dot{\lambda}_2 \biggr[ \biggl( \frac{1}{h_2} \frac{\partial h_2}{\partial\lambda_1} \biggr) \dot{\lambda}_1 + \biggl( \frac{1}{h_2} \frac{\partial h_2}{\partial\lambda_2} \biggr) \dot{\lambda}_2 \biggl] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="2">
<math>
\biggl( h_1 \frac{\partial h_1}{\partial\lambda_2} \biggr) \dot{\lambda}_1^2 + 
\biggl( h_2 \frac{\partial h_2}{\partial\lambda_2} \biggr) \dot{\lambda}_2^2  
</math>
  </td>
</tr>
</table> 


===Binney and Tremaine Formalism===

We have also derived the second component of the equation of motion following the formalism outlined by Binney and Tremaine ([[Appendix/References#BT87|BT87]]).  Specifically, in our introductory discussion of the [[Appendix/Ramblings/T3Integrals|T3 Coordinate System]] our '''[[Appendix/Ramblings/T3Integrals#EOM.01|Equation EOM.01]]''' has the form,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{d(h_2 \dot{\lambda}_2)}{dt}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr) \frac{dh_2}{dt} .
</math>
  </td>
</tr>

</table>

To compare this with the form derived using the Christoffel symbol formalism, we need to multiply through by <math>h_2</math> and bring the scale factor inside the time-derivative on the left-hand-side.

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{d(h_2^2 \dot{\lambda}_2)}{dt}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="3">
<math>
\biggl[ \biggl(\frac{h_2 \lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr)  + (h_2 \dot{\lambda}_2) \biggr]\frac{dh_2}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{h_2 \lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr)  + (h_2 \dot{\lambda}_2) \biggr]\biggl[ \frac{\partial h_2}{\partial\lambda_1} \dot{\lambda}_1 +  \frac{\partial h_2}{\partial\lambda_2} \dot{\lambda}_2 \biggr] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{h_2 \lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr)  + (h_2 \dot{\lambda}_2) \biggr]\biggl[ - \frac{\lambda_2}{\lambda_1} \dot{\lambda}_1 +  \dot{\lambda}_2 \biggr] \frac{\partial h_2}{\partial\lambda_2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="1">
<math>
\biggl[ \dot{\lambda}_2 + \frac{\lambda_2 }{\lambda_1} \dot{\lambda}_1  \biggr]\biggl[ \dot{\lambda}_2 - \frac{\lambda_2}{\lambda_1} \dot{\lambda}_1  \biggr] h_2 \frac{\partial h_2}{\partial\lambda_2} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="1">
<math>
\biggl[ \dot{\lambda}_2^2 - \biggl( \frac{\lambda_2 }{\lambda_1}\biggr)^2 \dot{\lambda}_1^2  \biggr] h_2 \frac{\partial h_2}{\partial\lambda_2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="1">
<math>
\biggl( h_2 \frac{\partial h_2}{\partial\lambda_2}\biggr) \dot{\lambda}_2^2 - \biggl[\frac{h_2 \lambda_2^2}{\lambda_1^2} \dot{\lambda}_1^2  \biggr] \biggl[- \frac{h_1 \lambda_1^2}{h_2 \lambda_2^2} \frac{\partial h_1}{\partial\lambda_2} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left" colspan="1">
<math>
\biggl( h_2 \frac{\partial h_2}{\partial\lambda_2}\biggr) \dot{\lambda}_2^2 + \biggl( h_1 \frac{\partial h_1}{\partial\lambda_2}\biggr) \dot{\lambda}_1^2
</math>
  </td>
</tr>
</table>

===Summary===
So we see that, indeed, the two formalisms produce identical forms of the equation of motion.


==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>
  </td>
</tr>

<tr>
  <td align="left">
<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="right">
<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>


<table border="1" cellpadding="10" width="90%" align="center">
<tr>
  <td align="left">
<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.
  </td>
</tr>
<tr><td>
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?

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


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>

==Conserved Quantity==

Let's cut to the chase.  As shown on the page describing the characteristic vector approach, I can write down the third conserved quantity right now--just not in closed form.  Assuming there's no potential variation in the direction of <math>\lambda_2</math>, it is

<div align="center">
<math>
m{h_2}^2 \dot{\lambda_2} \exp \left\{ - \int \frac{{h_k}^2}{{h_2}^2} \Gamma^k_{2j} \frac{\dot{\lambda_j} \dot{\lambda_k}}{\dot{\lambda_2}} \ dt \right\} .
</math>
</div>

In the case of T3 coordinates, this becomes more specific.

<div align="center">
<math>
m {h_2}^2 \dot{\lambda_2} \exp \left\{ - \int 2 {\lambda_1}^2 \ell^4 \left( \frac{\lambda_2 \dot{\lambda_1}^2}{\dot{\lambda_2}} - \frac{\dot{\lambda_2}{\lambda_1}^2}{\lambda_2} \right) dt \right\}
</math>
</div>

Although this is not all that useful in an analytic sense until we can integrate it, I wonder if it can be a guide to building a more accurate numerical model.  Certainly this function can be integrated numerically, and that's got to be useful somehow...

==Thoughts on Integrating This Conserved Quantity==

The quantity appearing inside the parentheses has an interesting symmetry.  Each variable appearing without a dot in the first term appears in the same place with a dot in the second term, and vice versa.  Certainly there must be some differentiation rule that will allow us to express this quantity as a total time derivative.

On the other hand, the factor of <math>\dot{\lambda_2}</math> appearing in the denominator of the first term is troublesome.  I can't think of any differentiation rule that puts a derivative in the denominator.  Product rule, quotient rule, and chain rule all end up ''multiplying'' by derivatives.  So I wonder if there's some way to eliminate the <math>\dot{\lambda_2}</math> in favor of undotted variables.  This would require transforming the equation of motion for the <math>\dot{\lambda_2}</math> coordinate into a first-order equation.  Right now, the second-order equation reads

<div align="center">
<math>
\ddot{\lambda_2} + \frac{\dot{h_2}}{h_2} \dot{\lambda_2} - \frac{\dot{\lambda_1} \dot{h_2}}{\lambda_1 h_2} \lambda_2 = 0 .
</math>
</div>

The first step in reducing this to a first-order equation is to perform a transformation of variables that eliminates that <math>\dot{\lambda_2}</math> term.  I have successfully accomplished this.  By defining <math>b \equiv {h_2}^{1/2} \lambda_2</math>, the equation can be written:

<div align="center">
<math>
\ddot{b} + \left( \tfrac{1}{4} \frac{{\dot{h_2}}^2}{h_2} - \tfrac{1}{2} \frac{\ddot{h_2}}{h_2} - \frac{\dot{\lambda_1} \dot{h_2}}{\lambda_1 h_2} \right) b = 0 .
</math>
</div>

&nbsp;<br />
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