User:Tohline/SSC/VariationalPrinciple - Revision history

Tohline: /* See Also */

2019-06-29T00:32:03Z

Tohline: /* See Also */

2019-06-29T00:22:20Z

Tohline: /* Ledoux & Pekeris Approach */

2017-06-17T17:00:51Z

Ledoux & Pekeris Approach

← Older revision		Revision as of 17:00, 17 June 2017
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	\, .</math>		\, .</math>
	</div>		</div>
	(~~As Ledoux & Pekeris point out immediately after providing~~ this ~~definition~~, ~~for a sphere~~ <math>~I</math> is ~~equal~~ to ~~<math>~\tfrac{2}{3}I_i</math>, where <math>~I_i</math> is~~ the moment of inertia ~~about one diameter~~ of the ~~sphere, as defined, for example,~~ [[User:Tohline/VE#~~MomentOfInertia~~\|~~here~~]].)		(Note that, defined in this way, <math>~I</math> is the same as [[User:Tohline/VE#Standard_Presentation_.5Bthe_Virial_of_Clausius_.281870.29.5D\|what we have referred to elsewhere]] as the ''scalar moment of inertia'', which is obtained by taking the trace of the [[User:Tohline/VE#MOItensor\|moment of inertia tensor]], <math>~I_{ij}</math>.)
	<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>		<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>

Tohline: /* Ledoux & Pekeris Approach */

2017-06-17T16:11:15Z

Ledoux & Pekeris Approach

← Older revision		Revision as of 16:11, 17 June 2017
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	\, .</math>		\, .</math>
	</div>		</div>
			(As Ledoux & Pekeris point out immediately after providing this definition, for a sphere <math>~I</math> is equal to <math>~\tfrac{2}{3}I_i</math>, where <math>~I_i</math> is the moment of inertia about one diameter of the sphere, as defined, for example, [[User:Tohline/VE#MomentOfInertia\|here]].)
	<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>		<span id="GoverningIntegral">After inserting these expressions, we have what will henceforth be referred to as the,</span>

Tohline: /* Plug in Known Marginally Unstable Solution */

2017-06-15T20:52:00Z

Plug in Known Marginally Unstable Solution

← Older revision		Revision as of 20:52, 15 June 2017
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	\frac{2^3}{3} \biggl[		\frac{2^3}{3} \biggl[
	\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)		\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
	\biggr]_0^{3} = \frac{2^3}{3} \biggl[ \frac{~~243~~}{64} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \biggr] \, .		\biggr]_0^{3} = \frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \biggr] \, .
	</math>		</math>
	</td>		</td>
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	\biggl\{ 15-\xi^2		\biggl\{ 15-\xi^2
	\biggr\}^2 d\xi		\biggr\}^2 d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-\frac{2^3}{3} \biggl[
			\frac{123\xi}{8(3+\xi^2)} - \frac{243\xi}{4(3+\xi^2)^2} + \frac{162\xi}{2(3+\xi^2)^3} + \frac{\xi}{2} - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
			\biggr]_0^{3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2^3}{3} \cdot \frac{5}{2^5} \biggl[ 2^2\cdot 3^{1 / 2} \pi - 3^3\biggr] = -\frac{2^3}{3} \biggl[ \frac{3^3\cdot 5}{2^5} - \frac{3^{1 / 2} \cdot 5\pi}{2^3} \bigg] \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~</math> RHS Total
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2^3}{3} \biggl[ \frac{3^5}{2^6} - \frac{3^3\cdot 5}{2^5} \biggr]
			= \frac{3^2}{2^3} \biggl[ 3^2 - 2\cdot 5 \biggr] = - \frac{3^2}{2^3} \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			LHS
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{2^2 5^2}{2\cdot 3} \biggl[ 3^3 \biggl(\frac{3}{3+3^2}\biggr)^{3} \frac{2^2}{5^2} \biggr]
			=
			- \frac{2^3 }{3} \biggl[\biggl(\frac{3}{2^2}\biggr)^{3} \biggr]
			=
			- \frac{3^2 }{2^3} \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
			Hence, the LHS = RHS.   <font size="+1" color="red"><b>Hooray!</b></font>
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Plug in Known Marginally Unstable Solution */

2017-06-15T20:24:19Z

Plug in Known Marginally Unstable Solution

← Older revision		Revision as of 20:24, 15 June 2017
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	\frac{2^3}{3} \biggl[		\frac{2^3}{3} \biggl[
	\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)		\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
	\biggr]_0^{3} = \frac{243}{64} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \, .		\biggr]_0^{3} = \frac{2^3}{3} \biggl[ \frac{243}{64} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \biggr] \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			RHS Term 2
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\int_0^{\tilde\xi} \xi^3 \theta^{n} \theta^'
			\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2 d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \int_0^{\tilde\xi} \xi^3 \biggl(\frac{3+\xi^2}{3}\biggr)^{-5 / 2} \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}
			\biggl\{ 4 - 2\frac{1}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}\biggl(\frac{3+\xi^2}{3}\biggr)^{5/2}
			\biggr\}^2 d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{1}{3}\int_0^{\tilde\xi} \biggl(\frac{3\xi}{3+\xi^2}\biggr)^{4}
			\biggl\{ 4 - \frac{2}{3}\biggl(\frac{3+\xi^2}{3}\biggr)
			\biggr\}^2 d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{2^3}{3}\int_0^{\tilde\xi} \frac{1}{2} \biggl(\frac{\xi}{3+\xi^2}\biggr)^{4}
			\biggl\{ 15-\xi^2
			\biggr\}^2 d\xi
	</math>		</math>
	</td>		</td>

Tohline: /* Plug in Known Marginally Unstable Solution */

2017-06-15T20:09:08Z

Plug in Known Marginally Unstable Solution

← Older revision		Revision as of 20:09, 15 June 2017
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	<math>~		<math>~
	\frac{2^3}{3} \int_0^{\tilde\xi} \frac{\xi^6 ~d\xi}{(3+\xi^2)^3}		\frac{2^3}{3} \int_0^{\tilde\xi} \frac{\xi^6 ~d\xi}{(3+\xi^2)^3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2^3}{3} \biggl[
			\frac{27\xi}{8(3+\xi^2)} - \frac{9\xi}{4(3+\xi^2)^2} + \xi - \biggl(\frac{3^{3/2}\cdot 5}{2^3} \biggr)\tan^{-1}\biggl(\frac{\xi}{3^{1 / 2}}\biggr)
			\biggr]_0^{3} = \frac{243}{64} - \frac{3^{1 / 2}\cdot 5\pi}{2^3} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Plug in Known Marginally Unstable Solution */

2017-06-15T19:34:17Z

Plug in Known Marginally Unstable Solution

← Older revision		Revision as of 19:34, 15 June 2017
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	2 \int_0^{\tilde\xi} \xi^2 \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2} \biggr]^{6} \biggl\{		2 \int_0^{\tilde\xi} \xi^2 \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2} \biggr]^{6} \biggl\{
	1 - \biggl[ \xi \biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]\frac{1}{\xi }\biggl(\frac{3+\xi^2}{3}\biggr)^{5 / 2}		1 - \biggl[ \xi \biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]\frac{1}{\xi }\biggl(\frac{3+\xi^2}{3}\biggr)^{5 / 2}
	+5 \biggl[ \frac{\xi^2}{3^2}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]~~^{-6}~~		+5 \biggl[ \frac{\xi^2}{3^2}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]
	\biggr\}^2 d\xi		\biggr\}^2 d\xi
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	2 \int_0^{\tilde\xi} ~~\xi^2 \biggl(~~\frac{3}{3+\xi^2}~~\biggr)^~~{~~3} \biggl\{~~		2 \int_0^{\tilde\xi} \frac{3^3 \xi^2}{(3+\xi^2)^3} \biggl\{
	~~1 - \biggl[ \xi \biggl~~(~~\frac{~~3+\xi^2~~}{3}\biggr~~)^{-3 ~~/ 2~~} \~~biggr]~~\~~frac~~{1~~}{\xi }~~\biggl(\frac{3+\xi^2}{3}\biggr)~~^{5 / 2}~~		1 - \biggl(\frac{3+\xi^2}{3}\biggr)
	+~~5 \biggl[~~ \frac{\xi^2}{3^2}~~\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]^{-6}~~		+ \frac{5\xi^2}{3^2}
	\biggr\}^2 d\xi		\biggr\}^2 d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2^3}{3} \int_0^{\tilde\xi} \frac{\xi^6 ~d\xi}{(3+\xi^2)^3}
	</math>		</math>
	</td>		</td>

Tohline: /* Plug in Known Marginally Unstable Solution */

2017-06-15T19:20:06Z

Plug in Known Marginally Unstable Solution

← Older revision		Revision as of 19:20, 15 June 2017
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	</div>		</div>

	~~Again, as an example, let~~'s plug in the functions that are appropriate to n = 5 ~~polytropic configurations~~.
			<table border="1" align="center" width="80%" cellpadding="5">
			<tr><td align="left">
			Let's check to see whether the terms in this last expression balance out when we plug in the functions that are appropriate for the marginally unstable, n = 5 configuration, namely,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="left">
			<math>~\theta_5 = \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2}</math>,
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>~\frac{d\theta_5}{d\xi} = - \frac{\xi}{3}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2}</math>.
			</td>
			</tr>
			</table>
			</div>
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			RHS Term 1
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(n-3) \int_0^{\tilde\xi} \xi^2 \theta^{n+1}
			\biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2 d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2 \int_0^{\tilde\xi} \xi^2 \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{-1 / 2} \biggr]^{6} \biggl\{
			1 - \biggl[ \xi \biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]\frac{1}{\xi }\biggl(\frac{3+\xi^2}{3}\biggr)^{5 / 2}
			+5 \biggl[ \frac{\xi^2}{3^2}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]^{-6}
			\biggr\}^2 d\xi
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2 \int_0^{\tilde\xi} \xi^2 \biggl(\frac{3}{3+\xi^2}\biggr)^{3} \biggl\{
			1 - \biggl[ \xi \biggl(\frac{3+\xi^2}{3}\biggr)^{-3 / 2} \biggr]\frac{1}{\xi }\biggl(\frac{3+\xi^2}{3}\biggr)^{5 / 2}
			+5 \biggl[ \frac{\xi^2}{3^2}\biggl(\frac{3+\xi^2}{3}\biggr)^{-3} \biggr] \biggl[ \biggl(\frac{3+\xi^2}{3}\biggr)^{3} \biggr]^{-6}
			\biggr\}^2 d\xi
			</math>
			</td>
			</tr>
			</table>
			</div>

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

	=See Also=		=See Also=

Tohline: /* For All Polytropic Indexes */

2017-06-15T18:46:10Z

For All Polytropic Indexes

← Older revision		Revision as of 18:46, 15 June 2017
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	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~~~
	~~\frac{1}{\xi^2}\biggl[ \frac{2n}{3(n-1)} \biggr]^2~~
	~~\xi^2 x^2 \biggl[ \biggl( \frac{d\ln x}{d\ln \xi}\biggr)^2 + (n-3) \biggl( \frac{d\ln\theta}{d\ln\xi} \biggr) \biggr]~~
	~~</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~\xi^2 (n-3) \biggl[ \frac{3}{2n} \biggr]^2\biggl\{
	\biggl(\frac{3-n}{~~n-1~~}\biggr)^2 \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2		(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
	+ ~~(n-3)~~ \xi \biggl( \frac{ \theta^'}{\theta} \biggr)		+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
	\biggl[1 + ~~\biggl~~(~~\frac{~~n-3~~}{n-1}\biggr~~) \biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2		\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
			\biggr\}
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			</div>
			Hence, after setting <math>~\sigma^2 = 0</math>, the [[#PolytropeRelation\|above rearranged integral relation]] becomes,
			<div align="center">
			<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>~
	\frac{1}{~~\xi^2~~ (n-3)}\biggl[ ~~\frac{2n}~~{3} \~~biggr]^2~~		- \frac{2^2 n^2}{3(n-3)} \biggl[ {\tilde\xi}^3 {\tilde\theta}^{n+1} x_\mathrm{surface}^2 \biggr]
	\xi^~~2 x^2 \biggl[ \biggl( \frac~~{d\~~ln x}{d~~\~~ln \xi~~}~~\biggr)~~^2 + ~~(n-3)~~ \~~biggl( \frac~~{~~d\ln\theta~~}~~{d\ln\xi} \biggr)~~ \biggr]
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
			\int_0^{\tilde\xi} \xi^2 \theta^{n+1} \biggl\{
	(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2		(n-3) \biggl[ 1 + \frac{3\theta^'}{\xi \theta^{n}} + \frac{n (\theta^')^2}{\theta^{n+1}}\biggr]^2
	+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)		+\xi \biggl( \frac{ \theta^'}{\theta} \biggr)
	\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2		\biggl[(n-1) + (n-3)\biggl( \frac{\theta^'}{\xi \theta^{n}}\biggr) \biggr]^2
			\biggr\} d\xi
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>

			Again, as an example, let's plug in the functions that are appropriate to n = 5 polytropic configurations.

	=See Also=		=See Also=

← Older revision		Revision as of 00:32, 29 June 2019
Line 2,796:		Line 2,796:
	<font color="green">A systematic method is presented for deriving the Lagrangian governing the evolution of small perturbations of arbitrary flows of a self-gravitating perfect fluid. The method is applied to a differentially rotating stellar model; the result is a Lagrangian equivalent to that of [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)]. A sufficient condition for stability of rotating stars, derived from this Lagrangian, is simplified greatly by using as trial functions not the three components of the Lagrangian displacement vector, but three scalar functions … This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.		<font color="green">A systematic method is presented for deriving the Lagrangian governing the evolution of small perturbations of arbitrary flows of a self-gravitating perfect fluid. The method is applied to a differentially rotating stellar model; the result is a Lagrangian equivalent to that of [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)]. A sufficient condition for stability of rotating stars, derived from this Lagrangian, is simplified greatly by using as trial functions not the three components of the Lagrangian displacement vector, but three scalar functions … This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.

	… we examine the special cases of (i) axially symmetric perturbations of a rotating star (as treated by Chandrasekhar ~~and~~ Lebovitz 1968); and (ii) perturbations of a nonrotating star (as treated by Chandrasekhar and Lebovitz 1964). We find that the stability criteria for those cases can also be simplified …</font>		… we examine the special cases of (i) axially symmetric perturbations of a rotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1968ApJ...152..267C/abstract S. Chandrasekhar & N. R. Lebovitz 1968]); and (ii) perturbations of a nonrotating star (as treated by [https://ui.adsabs.harvard.edu/abs/1964ApJ...140.1517C/abstract Chandrasekhar and Lebovitz 1964)]. We find that the stability criteria for those cases can also be simplified …</font>
	</td></tr></table>		</td></tr></table>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

← Older revision		Revision as of 00:22, 29 June 2019
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	* [[User:Tohline/Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29\|Derivations that support this chapter's discussion of the Ledoux Variational Principle]]		* [[User:Tohline/Appendix/Ramblings/LedouxVariationalPrinciple#Ledoux.27s_Variational_Principle_.28Supporting_Derivations.29\|Derivations that support this chapter's discussion of the Ledoux Variational Principle]]
			* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies''
			<table border="0" align="center" width="100%" cellpadding="1"><tr>
			<td align="center" width="5%"> </td><td align="left">
			<font color="green">A variational principle of great power is derived. It is naturally adapted for computers, and may be used to determine the stability of any fluid flow including those in differentially-rotating, self-gravitating stars and galaxies. The method also provides a powerful theoretical tool for studying general properties of eigenfunctions, and the relationships between secular and ordinary stability. In particular we prove the anti-sprial theorem indicating that no stable (or steady( mode can have a spiral structure</font>.
			</td></tr></table>
			* [https://ui.adsabs.harvard.edu/abs/1972ApJS...24..319S/abstract B. F. Schutz, Jr. (1972)], ApJSuppl., 24, 319: ''Linear Pulsations and Stability of Differentially Rotating Stellar Models. I. Newtonian Analysis''
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			<font color="green">A systematic method is presented for deriving the Lagrangian governing the evolution of small perturbations of arbitrary flows of a self-gravitating perfect fluid. The method is applied to a differentially rotating stellar model; the result is a Lagrangian equivalent to that of [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)]. A sufficient condition for stability of rotating stars, derived from this Lagrangian, is simplified greatly by using as trial functions not the three components of the Lagrangian displacement vector, but three scalar functions … This change of variables saves one from integrating twice over the star to find the effect of the perturbed gravitational field.

			… we examine the special cases of (i) axially symmetric perturbations of a rotating star (as treated by Chandrasekhar and Lebovitz 1968); and (ii) perturbations of a nonrotating star (as treated by Chandrasekhar and Lebovitz 1964). We find that the stability criteria for those cases can also be simplified …</font>
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