User:Tohline/Appendix/Ramblings/T2Integrals - Revision history

Tohline: /* Review */ fix power of q in definition of \ell

2010-05-23T17:43:36Z

Review: fix power of q in definition of \ell

← Older revision		Revision as of 17:43, 23 May 2010
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	<tr>		<tr>
	<td align="left" colspan="7">		<td align="left" colspan="7">
	where,        <math>\ell \equiv (\varpi^2 + q^2 z^2)^{-1/2}</math>.		where,        <math>\ell \equiv (\varpi^2 + q^4 z^2)^{-1/2}</math>.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Review */ fix typo in position vector definition

2010-05-23T13:50:03Z

Review: fix typo in position vector definition

← Older revision		Revision as of 13:50, 23 May 2010
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	<td align="left">		<td align="left">
	<math>		<math>
	\hat{i}x + \hat{j}y + \hat{k}		\hat{i}x + \hat{j}y + \hat{k}z
	</math>		</math>
	</td>		</td>

Tohline: /* First Special Case (quadratic) */ Fix a few errors

2010-05-22T02:14:53Z

First Special Case (quadratic): Fix a few errors

← Older revision		Revision as of 02:14, 22 May 2010
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	<td align="left">		<td align="left">
	<math>		<math>
	\biggl( \frac{\chi_1}{\chi_2} \biggr)^2 \biggl[ \frac{\tanh^2\Zeta}{1+\tanh^2\Zeta} \biggr](1+\tanh^2\Zeta) ;		\biggl( \frac{\chi_1}{\chi_2} \biggr)^2 \biggl[ \frac{\tanh^2\Zeta}{1+\tanh^2\Zeta} \biggr](1-\tanh^2\Zeta) ;
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\sinh\~~zeta~~ ~(1 + \sinh^2\Zeta)^{1/2}		\sinh\Zeta ~(1 + \sinh^2\Zeta)^{1/2}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\cosh\~~zeta~~ ~(\cosh^2\Zeta - 1)^{1/2} .		\cosh\Zeta ~(\cosh^2\Zeta - 1)^{1/2} .
	</math>		</math>
	</td>		</td>

Tohline: /* First Special Case (quadratic) */ Small addition

2010-05-22T01:54:28Z

First Special Case (quadratic): Small addition

← Older revision		Revision as of 01:54, 22 May 2010
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	</table>		</table>

	Notice that, because it is expressible entirely in terms of <math>\Chi_q</math>, the variable <math>\Zeta</math> is a function only of the product of the two key coordinates. Hence, the scale factor <math>h_1</math> is only a function of the product of the coordinates while <math>h_2</math> depends on the ratio, as well as the product of the two coordinates.		Notice that, because it is expressible entirely in terms of <math>\Chi_q</math>, the variable <math>\Zeta</math> is a function only of the product of the two key coordinates. Hence, the scale factor <math>h_1</math> is only a function of the product of the coordinates while <math>h_2</math> depends on the ratio, as well as the product of the two coordinates. In this context, note that <math>\Zeta</math> can be derived from the product of the two key coordinates via one of the following two relations:


			<table border="0" align="center" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			\frac{\chi_1\chi_2}{BA}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\sinh\zeta ~(1 + \sinh^2\Zeta)^{1/2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\cosh\zeta ~(\cosh^2\Zeta - 1)^{1/2} .
			</math>
			</td>
			</tr>

			</table>

	=See Also=		=See Also=

Tohline: /* First Special Case (quadratic) */ Extend manipulation of EOM quantities

2010-05-22T01:46:13Z

First Special Case (quadratic): Extend manipulation of EOM quantities

Show changes

Tohline: Examine EOM in special quadratic case

2010-05-21T21:29:58Z

Examine EOM in special quadratic case

← Older revision		Revision as of 21:29, 21 May 2010
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	</math>		</math>
	</div>		</div>


			=First Special Case (quadratic)=
			As has been [[User:Tohline/Appendix/Ramblings/T1Coordinates#First_Special_Case_.28quadratic.29\|discussed in an accompanying chapter]], when <math>q^2=2</math> the product of <math>\chi_1</math> and <math>\chi_2</math> shows up as a key quantity when inverting the coordinate definitions. In particular, by defining,
			<div align="center">
			<math>\Chi_q \equiv \frac{2\chi_1 \chi_2}{AB} = \sinh(2\Zeta) ,</math>
			</div>
			and it companion,
			<div align="center">
			<math>\Upsilon_q \equiv \cosh(2\Zeta) = \sqrt{1 + \Chi_q^2} ,</math>
			</div>

			we can write,

			<div align="center">
			<math>
			\varpi^2 = \frac{A^2}{2\chi_2^2} \biggl[ \sqrt{1 + \Chi_q^2} - 1 \biggr] = \frac{1}{2} \biggl( \frac{A}{\chi_2} \biggr)^2 (\Upsilon - 1) ;
			</math>
			</div>


			<div align="center">
			<math>
			z^2 = \frac{A^2}{4 \chi_2^2}\biggl[ \frac{1}{2} \Chi_q^2 +1 - \sqrt{1 + \Chi_q^2}\biggr] = \frac{1}{8} \biggl( \frac{A}{\chi_2} \biggr)^2 (\Upsilon - 1)(\Upsilon - 1) ;
			</math>
			</div>


			<table align="center" border="0" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			\ell^{-2}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\frac{A^2}{2\chi_2^2} \biggl[ \Chi_q^2 + 1 - \sqrt{ 1 + \Chi_q^2} \biggr] = \frac{1}{2} \biggl( \frac{A}{\chi_2} \biggr)^2 (\Upsilon - 1) \Upsilon.
			</math>
			</td>
			</tr>
			</table>





	=See Also=		=See Also=

Tohline: /* Integrals of Motion in T2 Coordinates */ EOM

2010-05-21T03:03:26Z

Integrals of Motion in T2 Coordinates: EOM

← Older revision		Revision as of 03:03, 21 May 2010
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	</tr>		</tr>
	</table>		</table>

			If the potential is only a function of the first T2 coordinate, <math>\chi_1</math>, then the <math>2^\mathrm{nd}</math> component of the equation of motion states,

			<div align="center">
			<math>
			\frac{d}{dt}( h_2\dot{\chi}_2 ) + h_1 \dot{\chi}_1 \biggl[ \dot{\chi}_2 \frac{1}{h_1}\frac{\partial h_2}{\partial \chi_1} - \dot{\chi}_2 \frac{1}{h_2}\frac{\partial h_1}{\partial \chi_2} \biggr] = 0 .
			</math>
			</div>

			In other words,
			<div align="center">
			<math>
			\frac{d}{dt}( h_2\dot{\chi}_2 ) = q^2 h_1^2 h_2 \ell^2 \dot{\chi}_1 \chi_2^2\frac{d}{dt}\biggl(\frac{\chi_1}{\chi_2}\biggr) .
			</math>
			</div>

			And the time derivative of a quantity that resembles the traditional angular momentum is,
			<div align="center">
			<math>
			\frac{d}{dt}( h_2^2 \dot{\chi}_2 ) = \frac{q^2 \ell^2 h_2^2}{\chi_2} \biggl\{ (h_1 \chi_2 \dot{\chi}_1)^2 - q^2 \ell^2 z^2 \biggl[(\varpi^2 + q^2 z^2) + (\varpi^2 + q^4 z^2) \biggr]\dot{\chi}_2^2 \biggr\} .
			</math>
			</div>

	=See Also=		=See Also=

Tohline: /* Integrals of Motion in T2 Coordinates */ Additional differential relations

2010-05-21T02:46:07Z

Integrals of Motion in T2 Coordinates: Additional differential relations

Show changes

Tohline: Begin discussion of T2 coordinates integral of motion

2010-05-20T02:21:27Z

Begin discussion of T2 coordinates integral of motion

New page

{{LSU_HBook_header}}

=Integrals of Motion in T2 Coordinates=
Motivated by the HNM82 derivation, in [[User:Tohline/Appendix/Ramblings/T1Coordinates#T2_Coordinates|an accompanying chapter]] we have introduced a new ''T2 Coordinate System'' and have outlined a few of its properties. Here we explore whether an analytic prescription of the <math>3^\mathrm{rd}</math> integral of motion can be formulated in a potential that conforms to this set of coordinates.

==Review==
We begin by summarizing properties of the T2 Coordinate System that we have derived earlier. By defining the dimensionless angle,
<div align="center">
<math>
\Zeta \equiv \sinh^{-1}\biggl( \frac{qz}{\varpi} \biggr) ,
</math>
</div>
the two key "T2" coordinates can be written as,

<table align="center" border="0" cellpadding="2">
<tr>
<td align="right">
<math>
\chi_1
</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>B \varpi \cosh\Zeta</math>
</td>
<td align="center">
      and      
</td>
<td align="right">
<math>
\chi_2
</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\frac{A \sinh\Zeta}{ \varpi^{q^2-1}}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
B ( \varpi^2 + q^2z^2 )^{1/2} = qB \xi_1
</math>
</td>
<td align="center">
 
</td>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{Aqz}{\varpi^{q^2}} = Aq \biggl[\frac{1}{\tan\xi_2} \biggr]^{q^2}
</math>
</td>
</tr>
</table>

Here are a variety of 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>\chi_1</math>
</td>
<td align="center">
<math>
\biggl(\frac{B^2}{\chi_1}\biggr) x
</math>
</td>
<td align="center">
<math>
\biggl(\frac{B^2}{\chi_1}\biggr) y
</math>
</td>
<td align="center">
<math>
\biggl(\frac{B^2}{\chi_1}\biggr) q^2 z
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\chi_2</math>
</td>
<td align="center">
<math>
- \biggl( \frac{q^3 A z}{\varpi^{q^2+2}} \biggr) x
</math>
</td>
<td align="center">
<math>
- \biggl( \frac{q^3 A z}{\varpi^{q^2+2}} \biggr) y
</math>
</td>
<td align="center">
<math>
\frac{qA}{\varpi^{q^2}}
</math>
</td>
</tr>

<tr>
<td align="center">
<math>\chi_3</math>
</td>
<td align="center">
<math>
- \biggl( \frac{1}{\varpi^{2}} \biggr) y
</math>
</td>
<td align="center">
<math>
+ \biggl( \frac{1}{\varpi^{2}} \biggr) x
</math>
</td>
<td align="center">
<math>
0
</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\chi_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\chi_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\chi_1}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\chi_1^2}{B^4 (\varpi^2 + q^4 z^2)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\chi_1^2 \ell^2}{B^4}
</math>
</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\chi_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\chi_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\chi_2}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{z^2 \varpi^2 }{\chi_2^2 (\varpi^2 + q^4 z^2)}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{z^2 \varpi^2 \ell^2}{\chi_2^2}
</math>
</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\chi_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\chi_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\chi_3}{\partial z} \biggr)^2 \biggr]^{-1}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\varpi^2
</math>
</td>
<td align="center">
 
</td>
<td align="left">
 
</td>
</tr>

<tr>
<td align="left" colspan="7">
where,        <math>\ell \equiv (\varpi^2 + q^2 z^2)^{-1/2}</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{i}x + \hat{j}y + \hat{k}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\hat{e}_1 (h_1 \chi_1) + (1 - q^2) \hat{e}_2 (h_2 \chi_2) .
</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'']

 <br />
{{LSU_HBook_footer}}