User:Tohline/Appendix/Ramblings/T3Integrals - Revision history

Tohline: /* Orthogonality Condition */ Check various minors of the determinant a la MF53

2010-07-05T19:35:37Z

Orthogonality Condition: Check various minors of the determinant a la MF53

← Older revision		Revision as of 19:35, 5 July 2010
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	</tr>		</tr>
	</table>		</table>
	Hence, the key orthogonality condition defined by [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.02\|Equation DC.02]] is satisfied.		Hence, the key orthogonality condition defined by [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.02\|Equation DC.02]] is satisfied. MF53 also gives us relationships that should apply ''between'' the various direction cosines if the coordinate system is orthogonal. Let's check a few cases to see whether <math>\gamma_{mn} = M_{mn}</math>, where "<math>M_{mn}</math> is the minor of <math>\gamma_{mn}</math> in the determinant <math>\|\gamma_{mn}\|</math>":
			<div align="center">
			<math>
			M_{11} = \gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} = - (-\varpi\ell)\frac{x}{\varpi} = + \ell x .
			</math> <br /><br />

			<math>
			M_{12} = \gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} = -\varpi\ell \biggl(-\frac{y}{\varpi}\biggr) = +\ell y .
			</math> <br /><br />

			<math>
			M_{13} = \gamma_{21}\gamma_{32} - \gamma_{22}\gamma_{31} = \biggl( \frac{q^2 x z \ell}{\varpi} \biggr) \frac{x}{\varpi} - \biggl( \frac{q^2 y z \ell}{\varpi} \biggr) \biggl( -\frac{y}{\varpi} \biggr) = q^2 \ell z.
			</math> <br /><br />

			<math>
			M_{31} = \gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22} = \ell y (-\varpi\ell) - q^2 \ell z \biggl(\frac{q^2 yz\ell}{\varpi}\biggr) = - \frac{y \ell^2}{\varpi} \biggl( \varpi^2 + q^4 z^2 \biggr) = - \frac{y}{\varpi} .
			</math> <br /><br />

			<math>
			M_{33} = \gamma_{11}\gamma_{22} - \gamma_{12}\gamma_{21} = \ell x \biggr(\frac{q^2 y z \ell}{\varpi} \biggr) - \ell y \biggr(\frac{q^2 x z \ell}{\varpi} \biggr) = 0 .
			</math>
			</div>
			All of these beautifully obey the relationship, <math>\gamma_{mn} = M_{mn}</math>.

	===Position Vector===		===Position Vector===

Tohline: /* Direction Cosines */ Fill out table of direction cosines

2010-07-05T18:59:40Z

Direction Cosines: Fill out table of direction cosines

← Older revision		Revision as of 18:59, 5 July 2010
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	<font color="darkblue">Direction Cosines for T3 Coordinates</font><br/>		<font color="darkblue">Direction Cosines for T3 Coordinates</font><br/>
	<math>\gamma_{ni} = h_n\frac{\partial\lambda_n}{\partial x_i}</math>		<math>\gamma_{ni} = h_n\frac{\partial\lambda_n}{\partial x_i}</math>
			</td>
			</tr>

			<tr>
			<td align="center">

			</td>
			<td align="center" colspan="3">
			'''<math>i</math>'''
			</td>
			</tr>

			<tr>
			<td align="center" valign="middle" rowspan="3">
			'''<math>n</math>'''
			</td>
			<td align="center" colspan="1">
			<math>\ell x</math>
			</td>
			<td align="center" colspan="1">
			<math>\ell y</math>
			</td>
			<td align="center" colspan="1">
			<math>q^2 \ell z</math>
			</td>
			</tr>

			<tr>
			<td align="center" colspan="1">
			<math>\frac{q^2 x z \ell}{\varpi}</math>
			</td>
			<td align="center" colspan="1">
			<math>\frac{q^2 y z \ell}{\varpi}</math>
			</td>
			<td align="center" colspan="1">
			<math>- \varpi \ell</math>
			</td>
			</tr>

			<tr>
			<td align="center" colspan="1">
			<math>- \frac{y}{\varpi}</math>
			</td>
			<td align="center" colspan="1">
			<math>+\frac{x}{\varpi}</math>
			</td>
			<td align="center" colspan="1">
			<math>0</math>
			</td>
			</tr>

			<tr>
			<td align="center" colspan="4">
			where:
			<math>\ell \equiv (\varpi^2 + q^4 z^2)^{-1/2}</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Definition */ Begin typing in table of direction cosines

2010-07-05T18:37:15Z

Definition: Begin typing in table of direction cosines

← Older revision		Revision as of 18:37, 5 July 2010
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	<td align="left" colspan="7">		<td align="left" colspan="7">
	where,        <math>\ell \equiv (\varpi^2 + q^4 z^2)^{-1/2}</math>.		where,        <math>\ell \equiv (\varpi^2 + q^4 z^2)^{-1/2}</math>.
			</td>
			</tr>
			</table>

			===Direction Cosines===
			The following table contains expressions for the [[User:Tohline/Appendix/Ramblings/DirectionCosines#Basic_Definitions_and_Relations\|nine direction cosines]].

			<table border="1" align="center" cellpadding="8">
			<tr>
			<td align="center" colspan="4">
			<font color="darkblue">Direction Cosines for T3 Coordinates</font><br/>
			<math>\gamma_{ni} = h_n\frac{\partial\lambda_n}{\partial x_i}</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Orthogonality Condition */ Change reference/link from equation DC.01 to DC.02

2010-07-05T16:14:13Z

Orthogonality Condition: Change reference/link from equation DC.01 to DC.02

← Older revision		Revision as of 16:14, 5 July 2010
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	===Orthogonality Condition===		===Orthogonality Condition===
	Next, let's use the example orthogonality condition [[User:Tohline/Appendix/Ramblings/DirectionCosines#Orthogonality\|derived elsewhere in connection with our overview of direction cosines]]. Specifically, let's see if [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.01\|Equation DC.01]] is satisfied.		Next, let's use the example orthogonality condition [[User:Tohline/Appendix/Ramblings/DirectionCosines#Orthogonality\|derived elsewhere in connection with our overview of direction cosines]]. Specifically, let's see if [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.02\|Equation DC.02]] is satisfied.
	<table align="center" border="0" cellpadding="5">		<table align="center" border="0" cellpadding="5">
	<tr>		<tr>
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	</tr>		</tr>
	</table>		</table>
	Hence, the key orthogonality condition defined by [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.01\|Equation DC.01]] is satisfied.		Hence, the key orthogonality condition defined by [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.02\|Equation DC.02]] is satisfied.

	===Position Vector===		===Position Vector===

Tohline: /* Definition */ Check orthogonality of T3 coordinates

2010-07-05T00:33:14Z

Definition: Check orthogonality of T3 coordinates

← Older revision		Revision as of 00:33, 5 July 2010
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	</table>		</table>

			===Relevant Partial Derivatives===
	Here are some relevant partial derivatives:		Here are some relevant partial derivatives:

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	</table>		</table>

			===Scale Factors===
	The scale factors are,		The scale factors are,

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	</table>		</table>

			===Orthogonality Condition===
			Next, let's use the example orthogonality condition [[User:Tohline/Appendix/Ramblings/DirectionCosines#Orthogonality\|derived elsewhere in connection with our overview of direction cosines]]. Specifically, let's see if [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.01\|Equation DC.01]] is satisfied.
			<table align="center" border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			\frac{\partial\lambda_1}{\partial \varpi} \cdot \frac{\partial\lambda_2}{\partial \varpi} +
			\frac{\partial\lambda_1}{\partial z} \cdot \frac{\partial\lambda_2}{\partial z}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\frac{\varpi}{\lambda_1} \biggl[ \frac{q^2}{q^2-1} \left( \frac{\varpi}{qz} \right)^{1/(q^2-1)}
			\biggr] -
			\frac{q^2z}{\lambda_1} \biggl[ \frac{1}{q^2-1} \left( \frac{\varpi^{q^2}}{qz^{q^2}} \right)^{1/(q^2-1)}
			\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\frac{q^2}{(q^2-1)\lambda_1} \biggl\{ \varpi\biggl[ \left( \frac{\varpi}{qz} \right)^{1/(q^2-1)}
			\biggr] -
			z \biggl[ \left( \frac{\varpi^{q^2}}{qz^{q^2}} \right)^{1/(q^2-1)}
			\biggr] \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\frac{q^2}{(q^2-1)\lambda_1} \biggl[ q^{-1/(q^2-1)} \varpi^{1 + 1/(q^2-1)} z^{-1/(q^2-1)} - q^{-1/(q^2-1)} \varpi^{q^2/(q^2-1)} z^{1-q^2/(q^2-1)} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			0 .
			</math>
			</td>
			</tr>
			</table>
			Hence, the key orthogonality condition defined by [[User:Tohline/Appendix/Ramblings/DirectionCosines#DC.01\|Equation DC.01]] is satisfied.

			===Position Vector===
	The position vector is,		The position vector is,

Tohline: /* Definition */ Add extra expression to Joel's table of derivatives wrt Cartesian coordinates to make easier the comparison with Jay's table of derivatives wrt Cylindrical coordinates

2010-06-24T21:34:49Z

Definition: Add extra expression to Joel's table of derivatives wrt Cartesian coordinates to make easier the comparison with Jay's table of derivatives wrt Cylindrical coordinates

← Older revision		Revision as of 21:34, 24 June 2010
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	<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)		=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi^2} \biggr)
			</math><br />
			<math>
			=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi} \biggr)
	</math>		</math>
	</td>		</td>
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	<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)		=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{y}{\varpi^2} \biggr)
			</math><br />
			<math>
			=\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{y}{\varpi} \biggr)
	</math>		</math>
	</td>		</td>
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	<math>		<math>
	=- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \frac{1}{z}		=- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz} \biggr]^{1/(q^2-1)} \frac{1}{z}
			</math>
			<br />
			<math>
			=- \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2}}{qz^{q^2}} \biggr]^{1/(q^2-1)}
	</math>		</math>
	</td>		</td>

Tohline: /* Implications of Equation of Motion */ More manipulation of 1st component of EOM

2010-06-23T01:42:32Z

Implications of Equation of Motion: More manipulation of 1st component of EOM

← Older revision		Revision as of 01:42, 23 June 2010
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	</math><br /><br />		</math><br /><br />
	<math>		<math>
	\Rightarrow ~~~~~ \frac{d(h_1 \dot{\lambda}_1)}{dt} = A(h_2 \dot{\lambda}_2) - \frac{1}{h_1}\frac{\partial\Phi}{\partial\lambda_1} = -\frac{\lambda_2 ~~h_2~~ \dot{\lambda}_2~~}{h_1\lambda_1}~~\frac{dh_2}{dt} - \frac{1}{h_1}\frac{\partial\Phi}{\partial\lambda_1}		\Rightarrow ~~~~~ \frac{d(h_1 \dot{\lambda}_1)}{dt} = A(h_2 \dot{\lambda}_2) - \frac{1}{h_1}\frac{\partial\Phi}{\partial\lambda_1} = - \biggl[ \frac{h_2 \lambda_2 }{h_1\lambda_1} \biggr]\dot{\lambda}_2 \frac{dh_2}{dt} - \frac{1}{h_1}\frac{\partial\Phi}{\partial\lambda_1}
	</math>		</math>
	</div>		</div>


	==Logarithmic Derivatives of Scale Factors==		==Logarithmic Derivatives of Scale Factors==

Tohline: Play with first component of the EOM

2010-06-23T01:27:05Z

Play with first component of the EOM

← Older revision		Revision as of 01:27, 23 June 2010
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	<font color="green">'''I'm not yet quite sure what to do with all of this.'''</font>		<font color="green">'''I'm not yet quite sure what to do with all of this.'''</font>
	-->		-->

			Let's look at the <math>1^\mathrm{st}</math> component of the equation of motion.

			<div align="center">
			<math>
			\hat{e}_1 \cdot \frac{d\vec{v}}{dt} = \frac{d(h_1 \dot{\lambda}_1)}{dt} - A(h_2 \dot{\lambda}_2) = - \frac{1}{h_1}\frac{\partial\Phi}{\partial\lambda_1}
			</math><br /><br />
			<math>
			\Rightarrow ~~~~~ \frac{d(h_1 \dot{\lambda}_1)}{dt} = A(h_2 \dot{\lambda}_2) - \frac{1}{h_1}\frac{\partial\Phi}{\partial\lambda_1} = -\frac{\lambda_2 h_2 \dot{\lambda}_2}{h_1\lambda_1}\frac{dh_2}{dt} - \frac{1}{h_1}\frac{\partial\Phi}{\partial\lambda_1}
			</math>
			</div>


	==Logarithmic Derivatives of Scale Factors==		==Logarithmic Derivatives of Scale Factors==

Tohline: /* See Also */ Add pointers to the special quadratic case

2010-06-09T21:24:00Z

See Also: Add pointers to the special quadratic case

← Older revision		Revision as of 21:24, 9 June 2010
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	</td></tr>		</td></tr>
	</table>		</table>

			=Special Case (quadratic)=

			On a [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase\|separate page]] we examine the structure of the second component of the equation of motion in the special "quadratic" case of <math>q^2 = 2</math>. Jay Call is independently investigating this same special case [[User:Jaycall/T3 Coordinates/Special_Case\|on yet another wiki page]] so that we can ultimately compare and check each others' results.



	=See Also=		=See Also=

Tohline: /* Implications of Equation of Motion */ Try adding a couple of right-justified equation numbers

2010-06-06T20:54:16Z

Implications of Equation of Motion: Try adding a couple of right-justified equation numbers

← Older revision		Revision as of 20:54, 6 June 2010
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	<math>		<math>
	\Rightarrow ~~~~~ \frac{d(h_2 \dot{\lambda}_2)}{dt} = - A(h_1 \dot{\lambda}_1)		\Rightarrow ~~~~~ \frac{d(h_2 \dot{\lambda}_2)}{dt} = - A(h_1 \dot{\lambda}_1)
	</math><~~br /><br~~ />		</math>
	or, inserting the relation derived above for <math>A</math> in terms of <math>dh_2/dt</math> for T3 coordinates<br /><br />		</div>

			or, inserting the relation derived above for <math>A</math> in terms of <math>dh_2/dt</math> for T3 coordinates,

			<span id="EOM.01"><table align="right" border="1" cellpadding="10" width="10%">
			<tr><th><font color="darkblue">EOM.01</font></th></tr>
			</table></span>
			<div align="center">
	<math>		<math>
	\Rightarrow ~~~~~ \frac{d(h_2 \dot{\lambda}_2)}{dt} = \biggl(\frac{\lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr) \frac{dh_2}{dt}		\Rightarrow ~~~~~ \frac{d(h_2 \dot{\lambda}_2)}{dt} = \biggl(\frac{\lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr) \frac{dh_2}{dt}
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	\Rightarrow ~~~~~ \frac{d(h_2 \lambda_1 \dot{\lambda}_2)}{dt} = \dot{\lambda}_1 \frac{d(h_2 \lambda_2)}{dt} .		\Rightarrow ~~~~~ \frac{d(h_2 \lambda_1 \dot{\lambda}_2)}{dt} = \dot{\lambda}_1 \frac{d(h_2 \lambda_2)}{dt} .
	</math>		</math>
	<~~br /><br~~ />		</div>
	Or, equivalently (but perhaps more perversely),<br /><br />
			Or, equivalently (but perhaps more perversely),

			<span id="EOM.02"><table align="right" border="1" cellpadding="10" width="10%">
			<tr><th><font color="darkblue">EOM.02</font></th></tr>
			</table></span>
			<div align="center">
	<math>		<math>
	\frac{\dot{\lambda}_2}{\lambda_2} \biggl[ \frac{1}{h_2 \dot{\lambda}_2} \frac{d(h_2 \dot{\lambda}_2)}{dt} \biggr] = \frac{\dot{\lambda}_1}{\lambda_1} \biggl[ \frac{1}{h_2} \frac{dh_2}{dt} \biggr]		\frac{\dot{\lambda}_2}{\lambda_2} \biggl[ \frac{1}{h_2 \dot{\lambda}_2} \frac{d(h_2 \dot{\lambda}_2)}{dt} \biggr] = \frac{\dot{\lambda}_1}{\lambda_1} \biggl[ \frac{1}{h_2} \frac{dh_2}{dt} \biggr]
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	</div><br />		</div><br />

	<font color="red">'''Note that ~~one of these last few expressions~~ is equivalent to the <i>simplest form</i> of the conservation that we derived — actually, Jay Call derived it — back in January, 2009.'''</font> Specifically,		<font color="red">'''Note that Equation '''EOM.01''' is equivalent to the <i>simplest form</i> of the conservation that we derived — actually, Jay Call derived it — back in January, 2009.'''</font> Specifically,
	<table align="center" border="1" cellpadding="10" width="30%">		<table align="center" border="1" cellpadding="10" width="30%">
	<tr>		<tr>