Difference between revisions of "User:Tohline/Appendix/Ramblings/T3Integrals"

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(→‎Definition: Continue defining T3 coordinates)
(→‎Definition: Fix some errors)
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Here are some relevant partial derivatives (<font color="green">there may be a mistake in the derivation of the partials of <math>\lambda_2</math></font>):   
Here are some relevant partial derivatives:   


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</math><br />
</math><br />
<math>
<math>
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi} \biggr)
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi^2} \biggr)
</math>
</math>
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</math><br />
</math><br />
<math>
<math>
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)}  \biggl( \frac{y}{\varpi} \biggr)
=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)}  \biggl( \frac{y}{\varpi^2} \biggr)
</math>
</math>
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</math><br />
</math><br />
<math>
<math>
=- \frac{q}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{q^2/(q^2-1)}  
=- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \frac{1}{z}
</math>
</math>
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The scale factors are (<font color="green">there is a mistake in the derivation of <math>h_2</math></font>),
The scale factors are,


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<math>
<math>
\frac{\lambda_1^2}{(\varpi^2 + q^4 z^2)}
\lambda_1^2 \ell^2  
</math>
</math>
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<math>=</math>
&nbsp;
   </td>
   </td>
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<math>
&nbsp;
\lambda_1^2 \ell^2
</math>
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<math>
<math>
\frac{(q^2-1)}{q^2} [\sinh\Zeta]^{2q^2/(q^2-1)} \frac{\varpi^2 }{(\varpi^2 + q^4 z^2)}
(q^2-1)^2 \biggl(\frac{\varpi z \ell}{\lambda_2} \biggr)^2  
</math>
</math>
   </td>
   </td>
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   <td align="center">
<math>=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
&nbsp;
\frac{(q^2-1)}{q^2} \biggl[\frac{\varpi}{\lambda_2} \biggr]^{2q^2} \varpi^2 \ell^2
</math>
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The position vector is,
The position vector is,
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<math>
<math>
\hat{e}_1 (h_1 \chi_1) + (1 - q^2) \hat{e}_2 (h_2 \chi_2) .
\hat{e}_1 (h_1 \lambda_1) + \hat{e}_2 (h_2 \lambda_2) .
</math>
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==Other Potentially Useful Differential Relations==
==Other Potentially Useful Differential Relations==

Revision as of 13:44, 23 May 2010

Whitworth's (1981) Isothermal Free-Energy Surface
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Integrals of Motion in T3 Coordinates

Motivated by the HNM82 derivation, in an accompanying chapter we have introduced a new T2 Coordinate System and have outlined a few of its properties. Here we offer a modest redefinition of the second radial coordinate in an effort to bring even more symmetry to the definition of the position vector, <math>\vec{x}</math>.


Definition

By defining the dimensionless angle,

<math> \Zeta \equiv \sinh^{-1}\biggl( \frac{qz}{\varpi} \biggr) , </math>

the two key "T3" coordinates will be written as,

<math> \lambda_1 </math>

<math>\equiv</math>

<math>\varpi \cosh\Zeta = ( \varpi^2 + q^2z^2 )^{1/2}</math>

      and      

<math> \lambda_2 </math>

<math>\equiv</math>

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

Here are some relevant partial derivatives:

 

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

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

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

<math>\lambda_1</math>

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

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

<math> \frac{q^2}{\lambda_1} </math>

<math>\lambda_2</math>

<math> \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) x </math>
<math> =\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi^2} \biggr) </math>

<math> \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) y </math>
<math> =\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{y}{\varpi^2} \biggr) </math>

<math> - \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\zeta} \biggr]^{q^2/(q^2-1)} \frac{q}{\varpi^{q^2}} </math>
<math> =- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \frac{1}{z} </math>

<math>\lambda_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\lambda_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial z} \biggr)^2 \biggr]^{-1} </math>

<math>=</math>

<math> \lambda_1^2 \ell^2 </math>

 

 

<math>h_2^2</math>

<math>=</math>

<math> \biggl[ \biggl( \frac{\partial\lambda_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial z} \biggr)^2 \biggr]^{-1} </math>

<math>=</math>

<math> (q^2-1)^2 \biggl(\frac{\varpi z \ell}{\lambda_2} \biggr)^2 </math>

 

 

<math>h_3^2</math>

<math>=</math>

<math> \biggl[ \biggl( \frac{\partial\lambda_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_3}{\partial z} \biggr)^2 \biggr]^{-1} </math>

<math>=</math>

<math> \varpi^2 </math>

 

 

where,        <math>\ell \equiv (\varpi^2 + q^2 z^2)^{-1/2}</math>.


The position vector is,

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

<math>=</math>

<math> \hat{i}x + \hat{j}y + \hat{k} </math>

<math>=</math>

<math> \hat{e}_1 (h_1 \lambda_1) + \hat{e}_2 (h_2 \lambda_2) . </math>


See Also

 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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