User:Tohline/Apps/Blaes85SlimLimit - Revision history

Tohline: Make first effort to summarize

2016-05-06T01:02:20Z

Make first effort to summarize

← Older revision		Revision as of 01:02, 6 May 2016
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->

	=~~Stability~~ of PP Tori in the Slim Torus Limit=		=Oscillations of PP Tori in the Slim Torus Limit=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

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	This matches equation (3.9) of Blaes85. According to [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], this equation "<font color="green">… is a standard eigenvalue problem whose only solutions are the Jacobi polynomials …</font>"		This matches equation (3.9) of Blaes85. According to [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], this equation "<font color="green">… is a standard eigenvalue problem whose only solutions are the Jacobi polynomials …</font>"

	~~{{LSU_WorkInProgress}}~~

	==Solution==		==Solution==
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	Again, this precisely matches equation (3.12) from Blaes85, so we appear to be on the right track. Putting this all together, [[#Table2\|Table 2]] presents the first three eigenvectors that are relevant to the stability analysis of slim, PP tori.		Again, this precisely matches equation (3.12) from Blaes85, so we appear to be on the right track. Putting this all together, [[#Table2\|Table 2]] presents the first three eigenvectors that are relevant to the stability analysis of slim, PP tori.


			{{LSU_WorkInProgress}}



	<div align="center">		<div align="center">
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	</table>		</table>
	</div>		</div>

			==Summary==

			The spatial structure of the normalized enthalpy perturbation associated with each <math>~(j,k)</math> mode of oscillation in a slim PP torus is,

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\delta W_j^{(0)}(\eta,\theta)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\eta^{\|k\|} J_j^{n-1,\|k\|}(2\eta^2-1) \cdot \exp (ik\theta) \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			and the associated oscillation frequency is,

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{n}\biggl[2j^2+2jn + 2j\|k\| + n\|k\|\biggr] \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Because the oscillation frequency is always positive, all modes are dynamically stable. Because the meridional plane amplitude is everywhere real, the azimuthal distortion associated with each nonaxisymmetric mode, <math>~m</math>, is completely uninteresting.

	=See Also=		=See Also=

Tohline: /* Jacobi Differential Equation */ Finished developing \Upsilon_2 for Table 2

2016-05-05T23:09:12Z

Jacobi Differential Equation: Finished developing \Upsilon_2 for Table 2

Show changes

Tohline: /* Jacobi Differential Equation */ Temporarily insert copy of Table1

2016-05-05T21:16:27Z

Jacobi Differential Equation: Temporarily insert copy of Table1

← Older revision		Revision as of 21:16, 5 May 2016
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	</div>		</div>

	~~As an aside~~, let's plug this coordinate transformation into Blaes85 expression (3.9) and see whether it takes the form of the Sturm-Liouville ODE. Specifically, making the substitutions,		Following up on this clue, let's plug this coordinate transformation into Blaes85 expression (3.9) and see whether it takes the form of the Sturm-Liouville ODE. Specifically, making the substitutions,
	<div align="center">		<div align="center">
	<math>~y ~\rightarrow~\tfrac{1}{2}(x+1)</math>      and       <math>~dy = \tfrac{1}{2}dx \, ,</math>		<math>~y ~\rightarrow~\tfrac{1}{2}(x+1)</math>      and       <math>~dy = \tfrac{1}{2}dx \, ,</math>
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	</table>		</table>
	</div>		</div>
	which is precisely in the standard Sturm-Liouville format. The j<sup>th</sup> eigenfunction should therefore be, <math>~J_j^{\alpha,\beta}(2\eta^2-1)</math>, and the associated eigenfrequency should be,		which is precisely in the standard Sturm-Liouville format. The j<sup>th</sup> eigenfunction should therefore be, <math>~\Upsilon_j = J_j^{\alpha,\beta}(2\eta^2-1)</math>, and the associated eigenfrequency should be,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	~~Again, this precisely matches equation (3.12) from Blaes85, so we appear to be on the right track. This time, we know that, for each "j" the corresponding eigenfunction is,~~
	~~<div align="center">~~
	~~<math>~J_j^{n-1,\|k\|}(2\eta^2-1) \, .</math>~~
	~~</div>~~

			Again, this precisely matches equation (3.12) from Blaes85, so we appear to be on the right track. Putting this all together, Table 2 presents the first three eigenvectors that are relevant to the stability analysis of slim, PP tori.

			<table align="center" border="1" cellpadding="5">
			<tr>
			<th align="center" colspan="3"><font size="+1">Table 1: Example Jacobi Polynomials</font></th>
			</tr>
			<tr>
			<td align="center"><math>~j</math></td>
			<td align="center"><math>~J_j^{\alpha,\beta}(x)</math></td>
			<td align="center"><math>~\lambda_j^{\alpha,\beta}</math></td>
			</tr>
			<tr>
			<td align="center">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~1</math>
			</td>
			<td align="center">
			<math>~0</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			<math>~1</math>
			</td>
			<td align="center">
			<math>~\tfrac{1}{2}(\alpha+\beta+2)x + \tfrac{1}{2}(\alpha-\beta)</math>
			</td>
			<td align="center">
			<math>~(\alpha+\beta+2)</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			<math>~2</math>
			</td>
			<td align="center">
			<math>~
			\tfrac{1}{8}(12+7\alpha + \alpha^2 + 7\beta+\beta^2+ 2\alpha\beta ) x^2
			+ \tfrac{1}{4}(3\alpha + \alpha^2 - 3\beta-\beta^2) x
			+ \tfrac{1}{8}(-4 - \alpha + \alpha^2-\beta + \beta^2 - 2\alpha\beta)
			</math>
			</td>
			<td align="center">
			<math>~2(\alpha+\beta+3)</math>
			</td>
			</tr>
			<tr>
			<td align="left" colspan="3">See also, eqs. (35)-(37) of [http://mathworld.wolfram.com/JacobiPolynomial.html Wolfram MathWorld].
			</tr>
			</table>

			<div align="center" id="Table2">
	<table align="center" border="1" cellpadding="5">		<table align="center" border="1" cellpadding="5">
	<tr>		<tr>
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	</tr>		</tr>
	</table>		</table>
			</div>

	=See Also=		=See Also=

Tohline: /* Singular Sturm-Liouville Problem */

2016-05-05T21:13:42Z

Singular Sturm-Liouville Problem

← Older revision		Revision as of 21:13, 5 May 2016
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	Table 1 presents this "j=2" eigenfunction — along with its corresponding eigenfrequency — as well as the eigenfunctions and eigenfrequencies for "j=0" and "j=1."		Table 1 presents this "j=2" eigenfunction — along with its corresponding eigenfrequency — as well as the eigenfunctions and eigenfrequencies for "j=0" and "j=1."
			<div align="center" id="Table1">
	<table align="center" border="1" cellpadding="5">		<table align="center" border="1" cellpadding="5">
	<tr>		<tr>
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	</tr>		</tr>
	</table>		</table>
			</div>

	Okay, so now I understand how to identify the complete set of eigenvectors for a stability problem in which the governing ODE takes the form of the singular Sturm-Liouville equation. But, with the recognition that the Blaes85 ODE does not have precisely the same form as the singular Sturm-Liouville equation, it is unclear how we will win by pursuing this specific line of investigation.		Okay, so now I understand how to identify the complete set of eigenvectors for a stability problem in which the governing ODE takes the form of the singular Sturm-Liouville equation. But, with the recognition that the Blaes85 ODE does not have precisely the same form as the singular Sturm-Liouville equation, it is unclear how we will win by pursuing this specific line of investigation.

Tohline: /* Jacobi Differential Equation */

2016-05-05T17:37:50Z

Jacobi Differential Equation

← Older revision		Revision as of 17:37, 5 May 2016
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	</table>		</table>
	</div>		</div>
	we see that its form is identical to this so-called "Jacobi's equation." The ~~association~~ is complete if we make the following parameter associations:		we see that its form is identical to this so-called "Jacobi's equation." The overlap is complete if we make the following parameter associations:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Singular Sturm-Liouville Problem */ Finish filling in Table 1 and related comments

2016-05-05T16:47:32Z

Singular Sturm-Liouville Problem: Finish filling in Table 1 and related comments

← Older revision		Revision as of 16:47, 5 May 2016
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	</table>		</table>
	</div>		</div>

	~~Table 1 provides a few examples.~~
	~~<table align="center" border="1" cellpadding="5">~~
	~~<tr>~~
	~~<th align="center" colspan="3"><font size="+1">Table 1: Example Jacobi Polynomials</font></th>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center"><math>~j</math></td>~~
	~~<td align="center"><math>~J_j^{\alpha,\beta}(x)</math></td>~~
	~~<td align="center"><math>~\lambda_j^{\alpha,\beta}</math></td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center">~~
	~~<math>~0</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~1</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~0</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center">~~
	~~<math>~1</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~\tfrac{1}{2}(\alpha+\beta+2)x + \tfrac{1}{2}(\alpha-\beta)</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~(\alpha+\beta+2)</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="left" colspan="3">See also, eqs. (35)-(37) of [http://mathworld.wolfram.com/JacobiPolynomial.html Wolfram MathWorld].~~
	~~</tr>~~
	~~</table>~~

	<!-- OLD NOTE		<!-- OLD NOTE
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{1}{2^3} \biggl\{		<math>~
			\frac{1}{2^3} \biggl\{
	[-4 - \alpha + \alpha^2-\beta + \beta^2 - 2\alpha\beta]		[-4 - \alpha + \alpha^2-\beta + \beta^2 - 2\alpha\beta]
	+ 2x[3\alpha + \alpha^2 - 3\beta-\beta^2)]		+ 2x[3\alpha + \alpha^2 - 3\beta-\beta^2)]
	+ x^2[12+7\alpha + \alpha^2 + 2\alpha\beta + 7\beta+\beta^2]		+ x^2[12+7\alpha + \alpha^2 + 2\alpha\beta + 7\beta+\beta^2]
	\biggr\} \, .		\biggr\}
	</math>		\, .</math>
	</td>		</td>
	</tr>		</tr>
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			Table 1 presents this "j=2" eigenfunction — along with its corresponding eigenfrequency — as well as the eigenfunctions and eigenfrequencies for "j=0" and "j=1."
			<table align="center" border="1" cellpadding="5">
			<tr>
			<th align="center" colspan="3"><font size="+1">Table 1: Example Jacobi Polynomials</font></th>
			</tr>
			<tr>
			<td align="center"><math>~j</math></td>
			<td align="center"><math>~J_j^{\alpha,\beta}(x)</math></td>
			<td align="center"><math>~\lambda_j^{\alpha,\beta}</math></td>
			</tr>
			<tr>
			<td align="center">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~1</math>
			</td>
			<td align="center">
			<math>~0</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			<math>~1</math>
			</td>
			<td align="center">
			<math>~\tfrac{1}{2}(\alpha+\beta+2)x + \tfrac{1}{2}(\alpha-\beta)</math>
			</td>
			<td align="center">
			<math>~(\alpha+\beta+2)</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			<math>~2</math>
			</td>
			<td align="center">
			<math>~
			\tfrac{1}{8}(12+7\alpha + \alpha^2 + 7\beta+\beta^2+ 2\alpha\beta ) x^2
			+ \tfrac{1}{4}(3\alpha + \alpha^2 - 3\beta-\beta^2) x
			+ \tfrac{1}{8}(-4 - \alpha + \alpha^2-\beta + \beta^2 - 2\alpha\beta)
			</math>
			</td>
			<td align="center">
			<math>~2(\alpha+\beta+3)</math>
			</td>
			</tr>
			<tr>
			<td align="left" colspan="3">See also, eqs. (35)-(37) of [http://mathworld.wolfram.com/JacobiPolynomial.html Wolfram MathWorld].
			</tr>
			</table>

	~~This~~, ~~combined~~ with the recognition that the Blaes85 ODE does not have precisely the same form as the singular Sturm-Liouville equation ~~makes me uncomfortable~~ pursuing this specific line of investigation.		Okay, so now I understand how to identify the complete set of eigenvectors for a stability problem in which the governing ODE takes the form of the singular Sturm-Liouville equation. But, with the recognition that the Blaes85 ODE does not have precisely the same form as the singular Sturm-Liouville equation, it is unclear how we will win by pursuing this specific line of investigation.

	===Jacobi Differential Equation===		===Jacobi Differential Equation===

Tohline: Work through derivation of J_2

2016-05-05T16:20:32Z

Work through derivation of J_2

Show changes

Tohline: /* Solution */ Begin filling in Table 2 examples

2016-05-05T00:39:09Z

Solution: Begin filling in Table 2 examples

← Older revision		Revision as of 00:39, 5 May 2016
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	</div>		</div>

			<table align="center" border="1" cellpadding="5">
			<tr>
			<th align="center" colspan="3"><font size="+1">Table 2: Example ''Slim'' PP-Tori Eigenvectors</font></th>
			</tr>
			<tr>
			<td align="center"><math>~j</math></td>
			<td align="center"><math>~\Upsilon_j = J_j^{n-1,\|k\|}(2\eta^2-1)</math></td>
			<td align="center"><math>~\biggl( \frac{\sigma}{\Omega_0} + m \biggr)_j^2</math></td>
			</tr>
			<tr>
			<td align="center">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~1</math>
			</td>
			<td align="center">
			<math>~0</math>
			</td>
			</tr>
			<tr>
			<td align="center">
			<math>~1</math>
			</td>
			<td align="center">
			<math>~(n + 1 + \|k\|)\eta^2 - (1 + \|k\|)</math>
			</td>
			<td align="center">
			<math>~\tfrac{1}{n}[2+2n+(2+n)\|k\|]</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: Finished transforming the governing ODE into the form of the singular Sturm-Liouville equation

2016-05-05T00:16:34Z

Finished transforming the governing ODE into the form of the singular Sturm-Liouville equation

Show changes

Tohline: /* Jacobi Differential Equation */ More work on solution

2016-05-04T21:17:17Z

Jacobi Differential Equation: More work on solution

← Older revision		Revision as of 21:17, 4 May 2016
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	This is identical to the eigenfrequency identified in eq. (3.12) of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]. So, we must be on the right track! Now I just need to figure out how to express the corresponding eigenfunction in terms of Jacobi polynomials.		This is identical to the eigenfrequency identified in eq. (3.12) of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]. So, we must be on the right track! Now I just need to figure out how to express the corresponding eigenfunction in terms of Jacobi polynomials.

			From expression 22.5.1 on p. 777 of A&S, we understand that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~G_j\biggl(\alpha+\beta + 1, \beta+1, \frac{x+1}{2}\biggr)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~P_j^{\alpha,\beta} (x) \biggl[ \frac{j! \Gamma(j + \alpha + \beta + 1)}{\Gamma(2j+\alpha+\beta+1)} \biggr] \, .</math>
			</td>
			</tr>
			</table>
			</div>
			In addition, we see from eq. (3.11) of Blaes85 that, to within a factor of a leading constant, the physically relevant eigenfunction is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Upsilon(y)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~A_{jk} G_j(\|k\|+n, \|k\|+1, \eta^2) \, .</math>
			</td>
			</tr>
			</table>
			</div>

			By association, we conclude from this that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\beta</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~\|k\| \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\alpha + \|k\|+1</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~\|k\|+n \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{x+1}{2}</math>
			</td>
			<td align="center">
			<math>~\leftrightarrow</math>
			</td>
			<td align="left">
			<math>~\eta^2 =y\, .</math>
			</td>
			</tr>
			</table>
			</div>
			From this third association, we conclude that the argument of the Jacobi polynomial should be,
			<div align="center">
			<math>x = 2\eta^2-1 \, .</math>
			</div>

	=See Also=		=See Also=