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=Integrals of Motion in T4 Coordinates=
[[User:Tohline/Appendix/Ramblings/T3Integrals|In an accompanying Wiki document]], we have derived the properties of an orthogonal, axisymmetric, ''T3'' coordinate system in which the first coordinate, <math>\lambda_1</math>, defines a family of concentric oblate-spheroidal surfaces whose (uniform) flattening is defined by a parameter <math>q \equiv R_\mathrm{eq}/Z_\mathrm{pole}</math>. In a [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase|separate, but related, Wiki document]], we attempt to derive the <math>3^\mathrm{rd}</math> isolating integral of motion for massless particles that move inside a flattened, axisymmetric potential whose equipotential surfaces align with <math>\lambda_1 = \mathrm{constant}</math> surfaces in the special (quadratic) case when <math>q^2 = 2</math>. While examining this special case, we noticed that, in T3 Coordinates, the <math>h_1</math> and <math>h_2</math> scale factors are only a function of the coordinate ratio <math>\lambda_1/\lambda_2</math>. This has led us to wonder whether it might be more fruitful to search for the <math>3^\mathrm{rd}</math> isolating integral using a coordinate system in which one of the coordinates is defined by this T3-coordinate ratio.

It is with this in mind that we explore the development of a new ''T4'' coordinate system. From the very beginning we will restrict the T4-coordinate definition to the special case of <math>q^2 = 2</math> because, at present, we think that the coordinate T3-coordinate ratio <math>\lambda_1/\lambda_2</math> is only interesting in the quadratic case. (See, for example, the polynomial root derived to complete the [[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29|T1-coordinate inversion for the cubic case]] <math>q^2=3</math>; it is another combination of the T3 coordinates that appears to be relevant in the cubic case.)

<div align="center">
<b><font color="red" size="+3">
STOP!
</font></b>

<b><font color="red" size="+1">
(7/06/2010)
</font></b>
</div>

<b><font color="red" size="+2">
As defined, below, this is ''not'' an orthogonal coordinate system.
</font></b>

==Definition==
In what follows, the coordinates <math>(\lambda_1,\lambda_2,\lambda_3)</math> refer to [[User:Tohline/Appendix/Ramblings/T3Integrals|T3 Coordinates]]. Let's define a set of orthogonal ''T4 Coordinates'' for the special (quadratic) case <math>q^2=2</math> such that,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\xi_1
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
(\lambda_1^2 + \lambda_2^2)^{1/2}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\varpi\biggl[1 + \sinh^2\Zeta + (\sinh\Zeta)^{2/(1-q^2)} \biggr]^{1/2}
</math>
</td>
<td align="center">
<math>
\xrightarrow{~~(q^2=2)~~}
</math>
</td>
<td align="left">
<math>
\varpi\biggl[1 + \sinh^2\Zeta + \frac{1}{\sinh^2\Zeta} \biggr]^{1/2} ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\xi_2
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\frac{\lambda_2}{\lambda_1}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(\sinh\Zeta)^{2/(1-q^2)}}{1+\sinh^2\Zeta} \biggr]^{1/2}
</math>
</td>
<td align="center">
<math>
\xrightarrow{~~(q^2=2)~~}
</math>
</td>
<td align="left">
<math>
\biggl[ \frac{1}{\sinh^2\Zeta(1+\sinh^2\Zeta)} \biggr]^{1/2} ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tan\xi_3
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\frac{y}{x} ,
</math>
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
</tr>
</table>

where,
<div align="center">
<math>
\sinh^2\Zeta \equiv \biggl(\frac{qz}{\varpi}\biggr)^2 ~~~~\xrightarrow{~~(q^2=2)~~}
~~~~ \frac{2z^2}{\varpi^2} .
</math>
</div>

The coordinate inversion — from <math>(\xi_1,\xi_2,\xi_3)</math> back to <math>(\lambda_1,\lambda_2,\lambda_3)</math> — is straightforward. Specifically,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\lambda_1
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\xi_1 \cos\biggl[ \tan^{-1}\xi_2 \biggr] ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\lambda_2
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\xi_1 \sin\biggl[ \tan^{-1}\xi_2 \biggr] ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\lambda_3
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\xi_3 .
</math>
</td>
</tr>
</table>

Here are some relevant partial derivatives:
<div align="center">
<math>
\frac{\partial\sinh^2\Zeta}{\partial\varpi} = -\frac{4z^2}{\varpi^3} ;
</math><br /><br />

<math>
\frac{\partial\sinh^2\Zeta}{\partial z} = + \frac{4z}{\varpi^2} .
</math>

</div>

Partial derivatives with respect to cylindrical coordinates are,

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center">
 
</td>
<td align="center">
<math>
\frac{\partial}{\partial \varpi}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial z}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial \phi}
</math>
</td>
</tr>

<tr>
<td align="center">
<math>{\xi_1}</math>
</td>
<td align="center">
<math>
\frac{\varpi}{\xi_1 z^2}\biggl(\varpi^2 + z^2 \biggr)
</math>
</td>
<td align="center">
<math>
\frac{1}{2\xi_1 z^3}\biggl(4z^4 - \varpi^4 \biggr)
</math>
</td>
<td align="center">
<math>
0
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\xi_2</math>
</td>
<td align="center">
<math>
\frac{2\xi_2^3 z^2}{\varpi^5}(\varpi^2 + 4z^2)
</math>
</td>
<td align="center">
<math>
- \frac{2\xi_2^3 z}{\varpi^4}(\varpi^2 + 4z^2)
</math>
</td>
<td align="center">
<math>
0
</math><br />
</td>
</tr>

<tr>
<td align="center">
<math>\xi_3</math>
</td>
<td align="center">
<math>
0
</math>
</td>
<td align="center">
<math>
0
</math>
</td>
<td align="center">
<math>
1
</math>
</td>
</tr>
</table>

Hence, the partials with respect to Cartesian coordinates are,

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center">
 
</td>
<td align="center">
<math>
\frac{\partial}{\partial x}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial y}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial z}
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\xi_1</math>
</td>
<td align="center">
<math>
\frac{x}{(1-q^2)\xi_1} \biggl[ 1 + \frac{q^4 z^2}{\varpi^2} - \frac{q^2 \xi_1^2}{\varpi^2} \biggr]
</math><br/><br/>
<math>\xrightarrow{(q^2=2)}~~~
\frac{x}{\xi_1 z^2} (\varpi^2 + z^2)</math>
</td>
<td align="center">
<math>
\frac{y}{(1-q^2)\xi_1} \biggl[ 1 + \frac{q^4 z^2}{\varpi^2} - \frac{q^2 \xi_1^2}{\varpi^2} \biggr]
</math><br/><br/>
<math>\xrightarrow{(q^2=2)}~~~
\frac{y}{\xi_1 z^2} (\varpi^2 + z^2)
</math>
</td>
<td align="center">
<math>
- \frac{\varpi^2}{(1-q^2)\xi_1 z} \biggl[ 1 + \frac{q^4 z^2}{\varpi^2} - \frac{\xi_1^2}{\varpi^2} \biggr]
</math><br/><br/>
<math>\xrightarrow{(q^2=2)}~~~
+ \frac{1}{2\xi_1 z^3}\biggl(4z^4 - \varpi^4 \biggr)
</math>

</td>
</tr>

<tr>
<td align="center">
<math>\xi_2</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\xi_3</math>
</td>
<td align="center">
<math>
-\frac{y}{\varpi^2}
</math>
</td>
<td align="center">
<math>
+\frac{x}{\varpi^2}
</math>
</td>
<td align="center">
<math>
0
</math>
</td>
</tr>
</table>

The scale factors are,

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>h_1^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{\partial\xi_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\xi_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{\partial\xi_1}{\partial \varpi} \biggr)^2 + \biggl( \frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\biggl[ \frac{4\xi_1^2 z^6 }{(\varpi^2 + 4z^2)(\varpi^6 + 4z^6)} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>h_2^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{\partial\xi_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\xi_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\xi_2}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{\partial\xi_2}{\partial \varpi} \biggr)^2 + \biggl( \frac{\partial\xi_2}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{\varpi^{10}}{4\xi_2^6 z^2} \biggl[ \frac{1}{(\varpi^2 + 4z^2)^2(\varpi^2 + z^2)} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>h_3^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \biggl( \frac{\partial\xi_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\xi_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\xi_3}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\varpi^2
</math>
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
</tr>

<tr>
<td align="left" colspan="7">
where,        <math>~~</math>.
</td>
</tr>
</table>


The position vector is,

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>\vec{x}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\hat\imath x + \hat\jmath y + \hat{k}z
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
~~
</math>
</td>
</tr>
</table>

=See Also=
* [http://www.phys.lsu.edu/astro/H_Book.current/Appendices/Integrals/integrals.of.motion.pdf Old discussion of ''Integrals of Motion'']

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Appendix/Ramblings/T4Integrals - Revision history

Jet53man: Created page with "__FORCETOC__ =Integrals of Motion in T4 Coordinates= In an accompanying Wiki document,..."

Jet53man: Created page with "FORCETOC =Integrals of Motion in T4 Coordinates= In an accompanying Wiki document,..."