Difference between revisions of "User:Tohline/Appendix/Ramblings/T4Integrals"

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(→‎Integrals of Motion in T4 Coordinates: Restrict discussion to the special quadratic case)
(→‎Integrals of Motion in T4 Coordinates: Abandon this discussion because the defined coordinates are not orthogonal)
 
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Line 6: Line 6:
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.)
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==
==Definition==
Line 38: Line 51:
   <td align="center">
   <td align="center">
<math>
<math>
\rightarrow^{q^2=2}
\xrightarrow{~~(q^2=2)~~}
</math>
</math>
   </td>
   </td>
Line 51: Line 64:
   <td align="right">
   <td align="right">
<math>
<math>
\tan\xi_2
\xi_2
</math>
</math>
   </td>
   </td>
Line 76: Line 89:
   <td align="center">
   <td align="center">
<math>
<math>
\rightarrow^{q^2=2}
\xrightarrow{~~(q^2=2)~~}
</math>
</math>
   </td>
   </td>
Line 120: Line 133:
<div align="center">
<div align="center">
<math>
<math>
\sinh\Zeta \equiv \frac{qz}{\varpi} \rightarrow^{q^2=2} \frac{2z^2}{\varpi^2} .
\sinh^2\Zeta \equiv \biggl(\frac{qz}{\varpi}\biggr)^2 ~~~~\xrightarrow{~~(q^2=2)~~}
~~~~ \frac{2z^2}{\varpi^2} .
</math>
</math>
</div>
</div>
Line 139: Line 153:
   <td align="left">
   <td align="left">
<math>
<math>
\xi_1 \cos\xi_2 ;
\xi_1 \cos\biggl[ \tan^{-1}\xi_2 \biggr] ;
</math>
</math>
   </td>
   </td>
Line 157: Line 171:
   <td align="left">
   <td align="left">
<math>
<math>
\xi_1 \sin\xi_2 ;
\xi_1 \sin\biggl[ \tan^{-1}\xi_2 \biggr] ;
</math>
</math>
   </td>
   </td>
Line 182: Line 196:


Here are some relevant partial derivatives:   
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">
<table align="center" border="1" cellpadding="5">
Line 190: Line 216:
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\partial}{\partial x}
\frac{\partial}{\partial \varpi}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\partial}{\partial y}
\frac{\partial}{\partial z}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\partial}{\partial z}
\frac{\partial}{\partial \phi}
</math>
</math>
   </td>
   </td>
Line 207: Line 233:
<tr>
<tr>
   <td align="center">
   <td align="center">
<math>\xi_1</math>
<math>{\xi_1}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<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]
\frac{\varpi}{\xi_1 z^2}\biggl(\varpi^2 + z^2 \biggr)
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<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]
\frac{1}{2\xi_1 z^3}\biggl(4z^4 - \varpi^4 \biggr)
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<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]
0
</math>
</math>
   </td>
   </td>
Line 232: Line 258:
   <td align="center">
   <td align="center">
<math>
<math>
~~
\frac{2\xi_2^3 z^2}{\varpi^5}(\varpi^2 + 4z^2)
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
~~
- \frac{2\xi_2^3 z}{\varpi^4}(\varpi^2 + 4z^2)
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
~~
0
</math>
</math><br />
   </td>
   </td>
</tr>
</tr>
Line 253: Line 279:
   <td align="center">
   <td align="center">
<math>
<math>
-\frac{y}{\varpi^2}
0
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
+\frac{x}{\varpi^2}
0
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
0
1
</math>
</math>
   </td>
   </td>
Line 269: Line 295:
</table>
</table>


<!--
Hence, the partials with respect to Cartesian coordinates are,
Alternatively, partials can be taken with respect to the cylindrical coordinates, <math>\varpi</math>, <math>z</math> and <math>\phi</math>.  (Incidentally, I have reversed the traditional order of the <math>\phi</math> and <math>z</math> coordinates in an attempt to parallelize structure between cylindrical and T3 coordinates since <math>\lambda_3 \equiv \phi</math>.)


<table align="center" border="1" cellpadding="5">
<table align="center" border="1" cellpadding="5">
Line 279: Line 304:
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\partial}{\partial \varpi}
\frac{\partial}{\partial x}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\partial}{\partial z}
\frac{\partial}{\partial y}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\partial}{\partial \phi}
\frac{\partial}{\partial z}
</math>
</math>
   </td>
   </td>
Line 296: Line 321:
<tr>
<tr>
   <td align="center">
   <td align="center">
<math>{\lambda_1}</math>
<math>\xi_1</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\varpi}{\lambda_1}
\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>
</math><br/><br/>
<math>\xrightarrow{(q^2=2)}~~~
\frac{x}{\xi_1 z^2} (\varpi^2 + z^2)</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{q^2 z}{\lambda_1}
\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>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
0
- \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>
</math>
   </td>
   </td>
</tr>
</tr>
Line 317: Line 351:
<tr>
<tr>
   <td align="center">
   <td align="center">
<math>\lambda_2</math>
<math>\xi_2</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{q^2}{q^2-1} \left( \frac{\varpi}{qz} \right)^{1/(q^2-1)}
~~
</math><br />
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
-\frac{1}{q^2-1} \left( \frac{\varpi^{q^2}}{qz^{q^2}} \right)^{1/(q^2-1)}
~~
</math><br />
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
0
~~
</math><br />
</math>
   </td>
   </td>
</tr>
</tr>
Line 338: Line 372:
<tr>
<tr>
   <td align="center">
   <td align="center">
<math>\lambda_3</math>
<math>\xi_3</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
0
-\frac{y}{\varpi^2}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
0
+\frac{x}{\varpi^2}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
1
0
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
-->


The inverted partials are


<table align="center" border="1" cellpadding="5">
The scale factors are,
 
<table align="center" border="0" cellpadding="5">
<tr>
<tr>
  <td align="right">
<math>h_1^2</math>
  </td>
   <td align="center">
   <td align="center">
&nbsp;
<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>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<math>
\frac{\partial}{\partial \xi_1}
\biggl[ \biggl( \frac{\partial\xi_1}{\partial \varpi} \biggr)^2 + \biggl( \frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
\frac{\partial}{\partial \xi_2}
=
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="left">
<math>
<math>
\frac{\partial}{\partial \xi_3}
\biggl[ \frac{4\xi_1^2 z^6 }{(\varpi^2 + 4z^2)(\varpi^6 + 4z^6)} \biggr]
</math>
</math>
   </td>
   </td>
Line 384: Line 429:


<tr>
<tr>
  <td align="right">
<math>h_2^2</math>
  </td>
   <td align="center">
   <td align="center">
<math>x</math>
<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>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<math>
~~
\biggl[ \biggl( \frac{\partial\xi_2}{\partial \varpi} \biggr)^2 + \biggl( \frac{\partial\xi_2}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
~~
=
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="left">
<math>
<math>
~~
\frac{\varpi^{10}}{4\xi_2^6 z^2} \biggl[ \frac{1}{(\varpi^2 + 4z^2)^2(\varpi^2 + z^2)} \biggr]
</math>
</math>
   </td>
   </td>
Line 405: Line 461:


<tr>
<tr>
  <td align="right">
<math>h_3^2</math>
  </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<math>
y
\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>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<math>
~~
\varpi^2
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
&nbsp;
~~
  </td>
</math>
  <td align="left">
&nbsp;
   </td>
   </td>
   <td align="center">
</tr>
<math>
 
~~
<tr>
</math>
   <td align="left" colspan="7">
where, &nbsp; &nbsp; &nbsp; &nbsp;<math>~~</math>.
   </td>
   </td>
</tr>
</tr>
</table>


<!--
For the record, the inverted partials are
<table align="center" border="1" cellpadding="5">
<tr>
<tr>
   <td align="center">
   <td align="center">
<math>
&nbsp;
z
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
~~
\frac{\partial}{\partial \xi_1}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
~~
\frac{\partial}{\partial \xi_2}
</math>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>
~~
\frac{\partial}{\partial \xi_3}
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>


The scale factors are,
<table align="center" border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="center">
<math>h_1^2</math>
<math>x</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<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>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<math>
~~
~~
Line 475: Line 535:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>
  </td>
~~
  <td align="left">
</math>
&nbsp;
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="center">
<math>h_2^2</math>
<math>
y
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<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>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<math>
~~
~~
Line 503: Line 558:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>
  </td>
~~
  <td align="left">
</math>
&nbsp;
   </td>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="center">
<math>h_3^2</math>
<math>
z
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<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>
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
<math>
~~
~~
Line 531: Line 581:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>
  </td>
~~
  <td align="left">
</math>
&nbsp;
  </td>
</tr>
 
<tr>
  <td align="left" colspan="7">
where, &nbsp; &nbsp; &nbsp; &nbsp;<math>~~</math>.
   </td>
   </td>
</tr>
</tr>
</table>
</table>
 
-->
 
The position vector is,
The position vector is,



Latest revision as of 16:12, 6 July 2010

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Integrals of Motion in T4 Coordinates

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 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 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.)

STOP!

(7/06/2010)

As defined, below, this is not an orthogonal coordinate system.

Definition

In what follows, the coordinates <math>(\lambda_1,\lambda_2,\lambda_3)</math> refer to T3 Coordinates. Let's define a set of orthogonal T4 Coordinates for the special (quadratic) case <math>q^2=2</math> such that,

<math> \xi_1 </math>

<math> \equiv </math>

<math> (\lambda_1^2 + \lambda_2^2)^{1/2} </math>

<math> = </math>

<math> \varpi\biggl[1 + \sinh^2\Zeta + (\sinh\Zeta)^{2/(1-q^2)} \biggr]^{1/2} </math>

<math> \xrightarrow{~~(q^2=2)~~} </math>

<math> \varpi\biggl[1 + \sinh^2\Zeta + \frac{1}{\sinh^2\Zeta} \biggr]^{1/2} ; </math>

<math> \xi_2 </math>

<math> \equiv </math>

<math> \frac{\lambda_2}{\lambda_1} </math>

<math> = </math>

<math> \biggl[ \frac{(\sinh\Zeta)^{2/(1-q^2)}}{1+\sinh^2\Zeta} \biggr]^{1/2} </math>

<math> \xrightarrow{~~(q^2=2)~~} </math>

<math> \biggl[ \frac{1}{\sinh^2\Zeta(1+\sinh^2\Zeta)} \biggr]^{1/2} ; </math>

<math> \tan\xi_3 </math>

<math> \equiv </math>

<math> \frac{y}{x} , </math>

 

 

 

 

where,

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

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,

<math> \lambda_1 </math>

<math> = </math>

<math> \xi_1 \cos\biggl[ \tan^{-1}\xi_2 \biggr] ; </math>

<math> \lambda_2 </math>

<math> = </math>

<math> \xi_1 \sin\biggl[ \tan^{-1}\xi_2 \biggr] ; </math>

<math> \lambda_3 </math>

<math> = </math>

<math> \xi_3 . </math>

Here are some relevant partial derivatives:

<math> \frac{\partial\sinh^2\Zeta}{\partial\varpi} = -\frac{4z^2}{\varpi^3} ; </math>

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

Partial derivatives with respect to cylindrical coordinates are,

 

<math> \frac{\partial}{\partial \varpi} </math>

<math> \frac{\partial}{\partial z} </math>

<math> \frac{\partial}{\partial \phi} </math>

<math>{\xi_1}</math>

<math> \frac{\varpi}{\xi_1 z^2}\biggl(\varpi^2 + z^2 \biggr) </math>

<math> \frac{1}{2\xi_1 z^3}\biggl(4z^4 - \varpi^4 \biggr) </math>

<math> 0 </math>

<math>\xi_2</math>

<math> \frac{2\xi_2^3 z^2}{\varpi^5}(\varpi^2 + 4z^2) </math>

<math> - \frac{2\xi_2^3 z}{\varpi^4}(\varpi^2 + 4z^2) </math>

<math> 0 </math>

<math>\xi_3</math>

<math> 0 </math>

<math> 0 </math>

<math> 1 </math>

Hence, the partials with respect to Cartesian coordinates are,

 

<math> \frac{\partial}{\partial x} </math>

<math> \frac{\partial}{\partial y} </math>

<math> \frac{\partial}{\partial z} </math>

<math>\xi_1</math>

<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>

<math>\xrightarrow{(q^2=2)}~~~ \frac{x}{\xi_1 z^2} (\varpi^2 + z^2)</math>

<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>

<math>\xrightarrow{(q^2=2)}~~~ \frac{y}{\xi_1 z^2} (\varpi^2 + z^2) </math>

<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>

<math>\xrightarrow{(q^2=2)}~~~ + \frac{1}{2\xi_1 z^3}\biggl(4z^4 - \varpi^4 \biggr) </math>

<math>\xi_2</math>

<math> ~~ </math>

<math> ~~ </math>

<math> ~~ </math>

<math>\xi_3</math>

<math> -\frac{y}{\varpi^2} </math>

<math> +\frac{x}{\varpi^2} </math>

<math> 0 </math>


The scale factors are,

<math>h_1^2</math>

<math>=</math>

<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>

<math>=</math>

<math> \biggl[ \biggl( \frac{\partial\xi_1}{\partial \varpi} \biggr)^2 + \biggl( \frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1} </math>

<math> = </math>

<math> \biggl[ \frac{4\xi_1^2 z^6 }{(\varpi^2 + 4z^2)(\varpi^6 + 4z^6)} \biggr] </math>

<math>h_2^2</math>

<math>=</math>

<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>

<math>=</math>

<math> \biggl[ \biggl( \frac{\partial\xi_2}{\partial \varpi} \biggr)^2 + \biggl( \frac{\partial\xi_2}{\partial z} \biggr)^2 \biggr]^{-1} </math>

<math> = </math>

<math> \frac{\varpi^{10}}{4\xi_2^6 z^2} \biggl[ \frac{1}{(\varpi^2 + 4z^2)^2(\varpi^2 + z^2)} \biggr] </math>

<math>h_3^2</math>

<math>=</math>

<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>

<math>=</math>

<math> \varpi^2 </math>

 

 

where,        <math>~~</math>.

The position vector is,

<math>\vec{x}</math>

<math>=</math>

<math> \hat\imath x + \hat\jmath y + \hat{k}z </math>

<math>=</math>

<math> ~~ </math>

See Also

 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation