Difference between revisions of "User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase"
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==Special Case (Quadratic)== | ==Special Case (Quadratic)== | ||
On one accompanying wiki page we have [[User:Tohline/Appendix/Ramblings/T3Integrals#Integras_of_Motion_in_T3_Coordinates|introduced T3 Coordinates]] and on another we have described how [[User:Tohline/Appendix/Ramblings/T3CharacteristicVector#Characteristic_Vector_for_T3_Coordinates|Jay Call's Characteristic Vector] applies to T3 Coordinates. Here we investigate the properties of our T3 Coordinate system in the special case when <math>q^2 = 2</math>; Jay Call is | On one accompanying wiki page we have [[User:Tohline/Appendix/Ramblings/T3Integrals#Integras_of_Motion_in_T3_Coordinates|introduced T3 Coordinates]] and on another we have described how [[User:Tohline/Appendix/Ramblings/T3CharacteristicVector#Characteristic_Vector_for_T3_Coordinates|Jay Call's Characteristic Vector]] applies to T3 Coordinates. Here we investigate the properties of our T3 Coordinate system in the special case when <math>q^2 = 2</math>; Jay Call's independent analysis is recorded on a [[User:Jaycall/T3_Coordinates/Special_Case|separate page]]. | ||
When <math>q^2=2</math>, the two key coordinates are: | |||
<table border="0" align="center" cellpadding="5"> | |||
<tr> | |||
<td align="right" colspan="2"> | |||
<math> | |||
\lambda_1 | |||
</math> | |||
</td> | |||
<td align="center"> | |||
<math> | |||
= | |||
</math> | |||
</td> | |||
<td align="left"> | |||
<math> | |||
\varpi \cosh\Zeta | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right" colspan="2"> | |||
<math> | |||
\lambda_2 | |||
</math> | |||
</td> | |||
<td align="center"> | |||
<math> | |||
= | |||
</math> | |||
</td> | |||
<td align="left"> | |||
<math> | |||
\frac{\varpi}{\sinh\Zeta} | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="left"> | |||
Note also: | |||
</td> | |||
<td align="right" colspan="1"> | |||
<math> | |||
2\frac{\lambda_1}{\lambda_2} | |||
</math> | |||
</td> | |||
<td align="center"> | |||
<math> | |||
= | |||
</math> | |||
</td> | |||
<td align="left"> | |||
<math> | |||
2 \sinh\Zeta \cosh\Zeta = \sinh(2\Zeta) | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | <br /> | ||
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Revision as of 22:28, 9 June 2010
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T3 Coordinates (continued)
Special Case (Quadratic)
On one accompanying wiki page we have introduced T3 Coordinates and on another we have described how Jay Call's Characteristic Vector applies to T3 Coordinates. Here we investigate the properties of our T3 Coordinate system in the special case when <math>q^2 = 2</math>; Jay Call's independent analysis is recorded on a separate page.
When <math>q^2=2</math>, the two key coordinates are:
|
<math> \lambda_1 </math> |
<math> = </math> |
<math> \varpi \cosh\Zeta </math> |
|
|
<math> \lambda_2 </math> |
<math> = </math> |
<math> \frac{\varpi}{\sinh\Zeta} </math> |
|
|
Note also: |
<math> 2\frac{\lambda_1}{\lambda_2} </math> |
<math> = </math> |
<math> 2 \sinh\Zeta \cosh\Zeta = \sinh(2\Zeta) </math> |
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© 2014 - 2021 by Joel E. Tohline |