User:Jaycall/T3 Coordinates/Special Case - Revision history

Jaycall: /* Conserved Quantity */ Improved form of integral slightly

2010-07-17T19:31:39Z

Conserved Quantity: Improved form of integral slightly

← Older revision		Revision as of 19:31, 17 July 2010
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	<div align="center">		<div align="center">
	<math>- \int ~~\frac{~~{\lambda_2}^2 \Lambda^2 \left( \Lambda - 1 \right)~~}{4}~~ \ \dot{\Lambda} \ dt = - \int ~~\frac{~~{\lambda_2}^2 \Lambda^2 \left( \Lambda - 1 \right)~~}{4}~~ \ d\Lambda </math> .		<math>-\tfrac{1}{4} \int {\lambda_2}^2 \Lambda^2 \left( \Lambda - 1 \right) \ \dot{\Lambda} \ dt = - \tfrac{1}{4} \int {\lambda_2}^2 \Lambda^2 \left( \Lambda - 1 \right) \ d\Lambda </math> .
	</div>		</div>

	We ought to be able to integrate this, right...? Maybe we could handle the pesky <math>{\lambda_2}^2</math> with integration by parts...		We ought to be able to integrate this, right...? Maybe we could handle the pesky <math>{\lambda_2}^2</math> with integration by parts...

Jaycall: /* Conserved Quantity */ Expressed in terms of \Lambda

2010-07-17T19:17:48Z

Conserved Quantity: Expressed in terms of \Lambda

← Older revision		Revision as of 19:17, 17 July 2010
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	<div align="center">		<div align="center">
	<math>		<math>
	m{h_2}^2 \dot{\lambda_2} \exp \int \left[ \left( 4 {\lambda_1}^2 + {\lambda_2}^2 - \lambda_2 \sqrt{4{\lambda_1}^2 + {\lambda_2}^2} \right) \left( \frac{{\lambda_1}^2 \dot{\lambda_2}}{\lambda_2} - \frac{\lambda_2 {\dot{\lambda_1}}^2}{\dot{\lambda_2}} \right) \right] dt		m{h_2}^2 \dot{\lambda_2} \exp \int \left[ \left( 4 {\lambda_1}^2 + {\lambda_2}^2 - \lambda_2 \sqrt{4{\lambda_1}^2 + {\lambda_2}^2} \right) \left( \frac{{\lambda_1}^2 \dot{\lambda_2}}{\lambda_2} - \frac{\lambda_2 {\dot{\lambda_1}}^2}{\dot{\lambda_2}} \right) \right] dt .
	</math>		</math>
	</div>		</div>

			The quantity in brackets needs to be integrated. In terms of <math>\Lambda</math>, it can be written

			<div align="center">
			<math>{\lambda_2}^2 \Lambda \left( \Lambda - 1 \right) \left[ \frac{\left( \Lambda^2 - 1 \right)^{1/2}}{2} \lambda_1 \dot{\lambda_2} - {\lambda_2}^2 \frac{\dot{\lambda_1}}{\dot{\lambda_2}} \frac{\dot{\lambda_1}}{\lambda_2} \right]</math> .
			</div>

			Notice that the thing in square brackets looks very closely related to <math>\dot{\Lambda}</math>. Could this be a hint? If only we could figure out what <math>\frac{\dot{\lambda_1}}{\dot{\lambda_2}}</math> is, maybe we could factor out the <math>\lambda_1 \dot{\lambda_2} - \frac{\dot{\lambda_1}}{\lambda_2}</math>, which appears in <math>\dot{\Lambda}</math>...

			If, by some miracle, it should turn out that <math>\frac{\left( \Lambda^2 - 1 \right)^{1/2}}{2} = {\lambda_2}^2 \frac{\dot{\lambda_1}}{\dot{\lambda_2}}</math>, factorization would be possible and our integral would read

			<div align="center">
			<math>- \int \frac{{\lambda_2}^2 \Lambda^2 \left( \Lambda - 1 \right)}{4} \ \dot{\Lambda} \ dt = - \int \frac{{\lambda_2}^2 \Lambda^2 \left( \Lambda - 1 \right)}{4} \ d\Lambda </math> .
			</div>

			We ought to be able to integrate this, right...? Maybe we could handle the pesky <math>{\lambda_2}^2</math> with integration by parts...

Jaycall: /* Coordinate Transformations */ Improved form of R^2 expression

2010-07-17T18:32:51Z

Coordinate Transformations: Improved form of R^2 expression

← Older revision		Revision as of 18:32, 17 July 2010
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>-\frac{{\lambda_2~~}^2~~}{2} + \lambda_2 \sqrt{{\lambda_1}^2 + {\lambda_2}^2/4} = \frac{2{\lambda_1}^2}{\Lambda+1}</math>		<math>\frac{\lambda_2}{2} \left( - \lambda_2 + \sqrt{4{\lambda_1}^2 + {\lambda_2}^2} \right) = \frac{2{\lambda_1}^2}{\Lambda+1}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{1}{2^{3/2}} \left[ -\lambda_2 + \sqrt{4 {\lambda_1}^2+{\lambda_2}^2} \right] = \frac{\lambda_2}{2^{3/2}} \left( \Lambda -1 \right)</math>		<math>\frac{1}{2^{3/2}} \left( -\lambda_2 + \sqrt{4 {\lambda_1}^2+{\lambda_2}^2} \right) = \frac{\lambda_2}{2^{3/2}} \left( \Lambda -1 \right)</math>
	</td>		</td>
	</tr>		</tr>

Jaycall: /* Coordinate Transformations */ Added derivatives of \Lambda

2010-07-17T18:24:23Z

Coordinate Transformations: Added derivatives of \Lambda

← Older revision		Revision as of 18:24, 17 July 2010
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	</table>		</table>

	where <math>\Lambda \equiv \left[ 1 + \left( \frac{2 \lambda_1}{\lambda_2} \right)^2 \right]^{1/2}</math>		where <math>\Lambda \equiv \left[ 1 + \left( \frac{2 \lambda_1}{\lambda_2} \right)^2 \right]^{1/2}</math> .

			From this definition of <math>\Lambda</math>, we can compute both its partials with respect to the T3 coordinates, and its total time derivative.

			<div align="center">
			<math>\frac{\partial \Lambda}{\partial \lambda_1} = \frac{4 \lambda_1 / {\lambda_2}^2}{\Lambda} = \frac{2 \left( \Lambda^2 - 1 \right)^{1/2}}{\Lambda} \frac{1}{\lambda_2} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\partial \Lambda}{\partial \lambda_2} = - \frac{4 {\lambda_1}^2 / \lambda_2}{\Lambda} = - \frac{2 \left( \Lambda^2 - 1 \right)^{1/2}}{\Lambda} \lambda_1</math>
			</div>

			<div align="center">
			<math>\dot{\Lambda} = \frac{2 \left( \Lambda^2 - 1 \right)^{1/2}}{\Lambda} \left( \frac{\dot{\lambda_1}}{\lambda_2} - \lambda_1 \dot{\lambda_2} \right)</math>
			</div>

	==Partials of the Coordinates==		==Partials of the Coordinates==

Jaycall: /* Partials of the Coordinates */ Added definitions in terms of \Lambda

2010-07-17T18:00:50Z

Partials of the Coordinates: Added definitions in terms of \Lambda

← Older revision		Revision as of 18:00, 17 July 2010
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	<td align="center">		<td align="center">
	<math>		<math>
	\frac{R}{\lambda_1}		\frac{R}{\lambda_1} = \left( \frac{2}{\Lambda + 1} \right)^{1/2}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>		<math>
	\frac{2z}{\lambda_1}		\frac{2z}{\lambda_1} = \left[ \frac{2 \left( \Lambda - 1 \right) }{\Lambda + 1} \right]^{1/2}
	</math>		</math>
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>		<math>
	\frac{2 \lambda_2}{R}		\frac{2 \lambda_2}{R} = \frac{2^{3/2}}{\left( \Lambda - 1 \right)^{1/2}}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>		<math>
	-\frac{\lambda_2}{z}		-\frac{\lambda_2}{z} = - \frac{2^{3/2}}{\Lambda - 1}
	</math>		</math>
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>		<math>
	R \ell^2 \lambda_1		R \ell^2 \lambda_1 = \frac{1}{\Lambda} \left( \frac{\Lambda + 1}{2} \right)^{1/2}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>		<math>
	2Rz^2 \ell^2 / \lambda_2		2Rz^2 \ell^2 / \lambda_2 = \frac{\left( \Lambda - 1 \right)^{3/2}}{2 \Lambda}
	</math>		</math>
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>		<math>
	2z \ell^2 \lambda_1		2z \ell^2 \lambda_1 = \frac{1}{\Lambda} \left( \frac{\Lambda^2 - 1}{2} \right)^{1/2}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>		<math>
	-R^2 z \ell^2 / \lambda_2		-R^2 z \ell^2 / \lambda_2 = - \frac{\Lambda - 1}{2^{3/2} \Lambda}
	</math>		</math>
	</td>		</td>
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	</table>		</table>

	where <math>\ell \equiv \left( R^2 + 4z^2 \right)^{-1/2}</math>.		where <math>\ell \equiv \left( R^2 + 4z^2 \right)^{-1/2} = \frac{1}{\lambda_1} \left( \frac{\Lambda + 1}{2 \Lambda} \right)^{1/2}</math>.

	==Scale Factors==		==Scale Factors==

Jaycall: /* Scale Factors */ Added definitions in terms of \Lambda

2010-07-17T17:17:53Z

Scale Factors: Added definitions in terms of \Lambda

← Older revision		Revision as of 17:17, 17 July 2010
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\lambda_1 \ell</math>		<math>\lambda_1 \ell = \left( \frac{\Lambda+1}{2 \Lambda} \right)^{1/2}</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>Rz \ell / \lambda_2</math>		<math>Rz \ell / \lambda_2 = \frac{\Lambda-1}{2 \left( 2 \Lambda \right)^{1/2}}</math>
	</td>		</td>
	</tr>		</tr>

Jaycall: /* Coordinate Transformations */ Added definitions in terms of \Lambda

2010-07-17T17:13:40Z

Coordinate Transformations: Added definitions in terms of \Lambda

← Older revision		Revision as of 17:13, 17 July 2010
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>-\frac{{\lambda_2}^2}{2} + \lambda_2 \sqrt{{\lambda_1}^2 + {\lambda_2}^2/4}</math>		<math>-\frac{{\lambda_2}^2}{2} + \lambda_2 \sqrt{{\lambda_1}^2 + {\lambda_2}^2/4} = \frac{2{\lambda_1}^2}{\Lambda+1}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{1}{2^{3/2}} \left[ -\lambda_2 + \sqrt{4 {\lambda_1}^2+{\lambda_2}^2} \right] </math>		<math>\frac{1}{2^{3/2}} \left[ -\lambda_2 + \sqrt{4 {\lambda_1}^2+{\lambda_2}^2} \right] = \frac{\lambda_2}{2^{3/2}} \left( \Lambda -1 \right)</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

			where <math>\Lambda \equiv \left[ 1 + \left( \frac{2 \lambda_1}{\lambda_2} \right)^2 \right]^{1/2}</math>

	==Partials of the Coordinates==		==Partials of the Coordinates==

Jaycall: Added headings

2010-07-17T16:49:49Z

Added headings

← Older revision		Revision as of 16:49, 17 July 2010
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			==Coordinate Transformations==

	If the special case <math>q^2=2</math> is considered, it is possible to invert the coordinate transformations in closed form. The coordinate transformations and their inversions become		If the special case <math>q^2=2</math> is considered, it is possible to invert the coordinate transformations in closed form. The coordinate transformations and their inversions become

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	</tr>		</tr>
	</table>		</table>

			==Partials of the Coordinates==

	Partial derivatives of each of the T3 coordinates taken with respect to each of the cylindrical coordinates are:		Partial derivatives of each of the T3 coordinates taken with respect to each of the cylindrical coordinates are:
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	where <math>\ell \equiv \left( R^2 + 4z^2 \right)^{-1/2}</math>.		where <math>\ell \equiv \left( R^2 + 4z^2 \right)^{-1/2}</math>.

			==Scale Factors==

	Furthermore, the scale factors become		Furthermore, the scale factors become
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	</tr>		</tr>
	</table>		</table>

			==Useful Relationships==

	In this special case, there are some additional useful relationships between various combinations of cylindrical variables and their T3 equivalents which can be written out.		In this special case, there are some additional useful relationships between various combinations of cylindrical variables and their T3 equivalents which can be written out.
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	</tr>		</tr>
	</table>		</table>

			==Additional Partials==

	Partials of <math>\ell</math> can be taken with respect to the coordinates of either system. They are:		Partials of <math>\ell</math> can be taken with respect to the coordinates of either system. They are:
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	</tr>		</tr>
	</table>		</table>

			==Conserved Quantity==

	The conserved quantity associated with the <math>\lambda_2</math> coordinate is		The conserved quantity associated with the <math>\lambda_2</math> coordinate is

Jaycall: Corrected coordinate transformation

2010-07-17T16:44:33Z

Corrected coordinate transformation

← Older revision		Revision as of 16:44, 17 July 2010
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>-\frac{{\lambda_2}^2}{2} + \sqrt~~{\frac~~{{\~~lambda_2~~}^2~~}{4}~~ + ~~{\lambda_1}^2~~ {\lambda_2}^2}</math>		<math>-\frac{{\lambda_2}^2}{2} + \lambda_2 \sqrt{{\lambda_1}^2 + {\lambda_2}^2/4}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>-\frac{~~\lambda_2~~}{2~~\sqrt~~{2}} + \sqrt~~{\frac~~{4 {\lambda_1}^2+{\lambda_2}^2}~~{8}}~~</math>		<math>\frac{1}{2^{3/2}} \left[ -\lambda_2 + \sqrt{4 {\lambda_1}^2+{\lambda_2}^2} \right] </math>
	</td>		</td>
	</tr>		</tr>

Jaycall: Added conserved quantity

2010-07-14T16:40:23Z

Added conserved quantity

← Older revision		Revision as of 16:40, 14 July 2010
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	</tr>		</tr>
	</table>		</table>

			The conserved quantity associated with the <math>\lambda_2</math> coordinate is

			<div align="center">
			<math>
			m{h_2}^2 \dot{\lambda_2} \exp \int \left[ \left( 4 {\lambda_1}^2 + {\lambda_2}^2 - \lambda_2 \sqrt{4{\lambda_1}^2 + {\lambda_2}^2} \right) \left( \frac{{\lambda_1}^2 \dot{\lambda_2}}{\lambda_2} - \frac{\lambda_2 {\dot{\lambda_1}}^2}{\dot{\lambda_2}} \right) \right] dt
			</math>
			</div>