User:Tohline/Appendix/Ramblings/T4Integrals - Revision history

Tohline: /* Integrals of Motion in T4 Coordinates */ Abandon this discussion because the defined coordinates are not orthogonal

2010-07-06T16:12:04Z

Integrals of Motion in T4 Coordinates: Abandon this discussion because the defined coordinates are not orthogonal

← Older revision		Revision as of 16:12, 6 July 2010
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==

Tohline: /* Definition */ Correct error in h_1 expression

2010-07-01T19:27:27Z

Definition: Correct error in h_1 expression

← Older revision		Revision as of 19:27, 1 July 2010
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	<td align="left">		<td align="left">
	<math>		<math>
	4\xi_1^2 z^6 ~~\biggl[ \frac~~{\varpi^6 + 4z^~~6}{~~\varpi^2 + 4z^2} \biggr]		\biggl[ \frac{4\xi_1^2 z^6 }{(\varpi^2 + 4z^2)(\varpi^6 + 4z^6)} \biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Definition */ Add a few more derived partial-derivative and scale-factor expressions

2010-07-01T16:54:19Z

Definition: Add a few more derived partial-derivative and scale-factor expressions

Show changes

Tohline: /* Definition */ Add cartesian z derivative

2010-06-30T21:29:06Z

Definition: Add cartesian z derivative

← Older revision		Revision as of 21:29, 30 June 2010
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	\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{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><br/><br/>		</math><br/><br/>
	<math>\xrightarrow{~~(q^2=2)~~}		<math>\xrightarrow{(q^2=2)}~~~
	\frac{x}{\xi_1 z^2} (\varpi^2 + z^2)</math>		\frac{x}{\xi_1 z^2} (\varpi^2 + z^2)</math>
	</td>		</td>
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	\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{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><br/><br/>
	<math>\xrightarrow{~~(q^2=2)~~}		<math>\xrightarrow{(q^2=2)}~~~
	\frac{y}{\xi_1 z^2} (\varpi^2 + z^2)		\frac{y}{\xi_1 z^2} (\varpi^2 + z^2)
	</math>		</math>
Line 328:		Line 328:
	<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]		- \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>

Tohline: /* Definition */ Start using

2010-06-30T21:19:55Z

Definition: Start using

← Older revision		Revision as of 21:19, 30 June 2010
Line 38:		Line 38:
	<td align="center">		<td align="center">
	<math>		<math>
	\~~overrightarrow~~{~~(q^2=2)~~}		\xrightarrow{~~(q^2=2)~~}
	</math>		</math>
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>		<math>
	\~~overrightarrow~~{~~(q^2=2)~~}		\xrightarrow{~~(q^2=2)~~}
	</math>		</math>
	</td>		</td>
Line 120:		Line 120:
	<div align="center">		<div align="center">
	<math>		<math>
	\sinh^2\Zeta \equiv \biggl(\frac{qz}{\varpi}\biggr)^2 ~~~~\~~overrightarrow~~{~~(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 313:		Line 314:
	\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{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><br/><br/>		</math><br/><br/>
	<math>\~~overrightarrow~~{~~(q^2=2)~~}\frac{x}{\xi_1 z^2} (\varpi^2 + z^2)</math>		<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{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{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>

Tohline: /* Definition */ Correct mistakes in early derivative derivations

2010-06-29T18:15:32Z

Definition: Correct mistakes in early derivative derivations

← Older revision		Revision as of 18:15, 29 June 2010
Line 120:		Line 120:
	<div align="center">		<div align="center">
	<math>		<math>
	\sinh^2\Zeta \equiv \frac{qz}{\varpi} ~~~~\overrightarrow{~~(q^2=2)~~}~~~~ \frac{2z^2}{\varpi^2} .		\sinh^2\Zeta \equiv \biggl(\frac{qz}{\varpi}\biggr)^2 ~~~~\overrightarrow{~~(q^2=2)~~}~~~~ \frac{2z^2}{\varpi^2} .
	</math>		</math>
	</div>		</div>
Line 188:		Line 188:

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

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	<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{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>\overrightarrow{~~(q^2=2)~~}\frac{x}{\xi_1 z^2} (\varpi^2 + z^2)</math>
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Definition */ Improve presentation and begin to lay out partial derivative expressions

2010-06-26T23:17:31Z

Definition: Improve presentation and begin to lay out partial derivative expressions

Show changes

Tohline: /* Integrals of Motion in T4 Coordinates */ Restrict discussion to the special quadratic case

2010-06-25T20:16:27Z

Integrals of Motion in T4 Coordinates: Restrict discussion to the special quadratic case

← Older revision		Revision as of 20:16, 25 June 2010
Line 2:		Line 2:

	=Integrals of Motion in T4 Coordinates=		=Integrals of Motion in T4 Coordinates=
	[[User:Tohline/Appendix/Ramblings/T3Integrals\|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 [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase\|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.		[[User:Tohline/Appendix/Ramblings/T3Integrals\|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 [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase\|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 [[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.)


	==Definition==		==Definition==
	In what follows, the coordinates <math>(\lambda_1,\lambda_2,\lambda_3)</math> refer to [[User:Tohline/Appendix/Ramblings/T3Integrals\|T3 Coordinates]]. Let's define a set of orthogonal ''T4 Coordinates'' such that,		In what follows, the coordinates <math>(\lambda_1,\lambda_2,\lambda_3)</math> refer to [[User:Tohline/Appendix/Ramblings/T3Integrals\|T3 Coordinates]]. Let's define a set of orthogonal ''T4 Coordinates'' for the special (quadratic) case <math>q^2=2</math> such that,
	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
	<tr>		<tr>
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	<td align="left">		<td align="left">
	<math>		<math>
	\varpi\biggl[1 + \sinh^2\Zeta + (\sinh\Zeta)^{2/(1-q^2)} \biggr]^{1/2} ;		\varpi\biggl[1 + \sinh^2\Zeta + (\sinh\Zeta)^{2/(1-q^2)} \biggr]^{1/2}
			</math>
			</td>
			<td align="center">
			<math>
			\rightarrow^{q^2=2}
			</math>
			</td>
			<td align="left">
			<math>
			\varpi\biggl[1 + \sinh^2\Zeta + \frac{1}{\sinh^2\Zeta} \biggr]^{1/2} ;
	</math>		</math>
	</td>		</td>
Line 59:		Line 71:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \frac{(\sinh\Zeta)^{2/(1-q^2)}}{1+\sinh^2\Zeta} \biggr]^{1/2} ;		\biggl[ \frac{(\sinh\Zeta)^{2/(1-q^2)}}{1+\sinh^2\Zeta} \biggr]^{1/2}
			</math>
			</td>
			<td align="center">
			<math>
			\rightarrow^{q^2=2}
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{1}{\sinh^2\Zeta(1+\sinh^2\Zeta)} \biggr]^{1/2} ;
	</math>		</math>
	</td>		</td>
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	\frac{y}{x} ,		\frac{y}{x} ,
	</math>		</math>
			</td>
			<td align="center">

			</td>
			<td align="left">

	</td>		</td>
	<td align="center">		<td align="center">
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	<div align="center">		<div align="center">
	<math>		<math>
	\sinh\Zeta \equiv \frac{qz}{\varpi} .		\sinh\Zeta \equiv \frac{qz}{\varpi} \rightarrow^{q^2=2} \frac{2z^2}{\varpi^2} .
	</math>		</math>
	</div>		</div>

Tohline: /* Definition */ Begin to fill out definitions and derivations

2010-06-23T22:01:51Z

Definition: Begin to fill out definitions and derivations

← Older revision		Revision as of 22:01, 23 June 2010
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	<td align="left">		<td align="left">
	<math>		<math>
	(\lambda_1^2 + \lambda_2^2)^{1/2} ;		(\lambda_1^2 + \lambda_2^2)^{1/2}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\varpi\biggl[1 + \sinh^2\Zeta + (\sinh\Zeta)^{2/(1-q^2)} \biggr]^{1/2} ;
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{\lambda_2}{\lambda_1} ;		\frac{\lambda_2}{\lambda_1}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{(\sinh\Zeta)^{2/(1-q^2)}}{1+\sinh^2\Zeta} \biggr]^{1/2} ;
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{y}{x} .		\frac{y}{x} ,
	</math>		</math>
			</td>
			<td align="center">

			</td>
			<td align="left">

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

			where,
			<div align="center">
			<math>
			\sinh\Zeta \equiv \frac{qz}{\varpi} .
			</math>
			</div>

	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,		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,
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	<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]
	</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]
	</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]
	</math>		</math>
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>		<math>
	~~		-\frac{y}{\varpi^2}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>		<math>
	~~		+\frac{x}{\varpi^2}
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>		<math>
	~~		0
	</math>		</math>
	</td>		</td>
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	-->		-->

	~~Ghe~~ inverted partials are		The inverted partials are

	<table align="center" border="1" cellpadding="5">		<table align="center" border="1" cellpadding="5">
Line 510:		Line 543:
	</tr>		</tr>
	</table>		</table>



	=See Also=		=See Also=

Tohline: Begin presentation of T4 coordinates

2010-06-23T19:20:51Z

Begin presentation of T4 coordinates

New page

{{LSU_HBook_header}}

=Integrals of Motion in T4 Coordinates=
[[User:Tohline/Appendix/Ramblings/T3Integrals|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 [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase|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.

==Definition==
In what follows, the coordinates <math>(\lambda_1,\lambda_2,\lambda_3)</math> refer to [[User:Tohline/Appendix/Ramblings/T3Integrals|T3 Coordinates]]. Let's define a set of orthogonal ''T4 Coordinates'' such that,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\xi_1
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
(\lambda_1^2 + \lambda_2^2)^{1/2} ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tan\xi_2
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\frac{\lambda_2}{\lambda_1} ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\tan\xi_3
</math>
</td>
<td align="center">
<math>
\equiv
</math>
</td>
<td align="left">
<math>
\frac{y}{x} .
</math>
</td>
</tr>
</table>

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,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right">
<math>
\lambda_1
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\xi_1 \cos\xi_2 ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\lambda_2
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\xi_1 \sin\xi_2 ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\lambda_3
</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\xi_3 .
</math>
</td>
</tr>
</table>

Here are some relevant partial derivatives:

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center">
 
</td>
<td align="center">
<math>
\frac{\partial}{\partial x}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial y}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial z}
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\xi_1</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\xi_2</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\xi_3</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
</tr>
</table>



Ghe inverted partials are

<table align="center" border="1" cellpadding="5">
<tr>
<td align="center">
 
</td>
<td align="center">
<math>
\frac{\partial}{\partial \xi_1}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial \xi_2}
</math>
</td>
<td align="center">
<math>
\frac{\partial}{\partial \xi_3}
</math>
</td>
</tr>

<tr>
<td align="center">
<math>x</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
</tr>

<tr>
<td align="center">
<math>
y
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
</tr>

<tr>
<td align="center">
<math>
z
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
<td align="center">
<math>
~~
</math>
</td>
</tr>
</table>

The scale factors are,

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>h_1^2</math>
</td>
<td align="center">
<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 align="center">
<math>=</math>
</td>
<td align="left">
<math>
~~
</math>
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
</tr>

<tr>
<td align="right">
<math>h_2^2</math>
</td>
<td align="center">
<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 align="center">
<math>=</math>
</td>
<td align="left">
<math>
~~
</math>
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
</tr>

<tr>
<td align="right">
<math>h_3^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<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>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
~~
</math>
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
</tr>

<tr>
<td align="left" colspan="7">
where,        <math>~~</math>.
</td>
</tr>
</table>

The position vector is,

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>\vec{x}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\hat\imath x + \hat\jmath y + \hat{k}z
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
~~
</math>
</td>
</tr>
</table>

=See Also=
* [http://www.phys.lsu.edu/astro/H_Book.current/Appendices/Integrals/integrals.of.motion.pdf Old discussion of ''Integrals of Motion'']

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