Editing SSC/VariationalPrinciple (section)

==Ledoux and Pekeris (1941)==
Historically, by the 1940s, the LAWE was a relatively familiar one to astrophysicists.  For example, the opening paragraph of a 1941 paper by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux &amp; Pekeris] (1941, ApJ, 94, 124), reads:
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Paragraph extracted from [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux &amp; C. L. Pekeris (1941)]<p></p>
"''Radial Pulsations of Stars''"<p></p>
ApJ, vol. 94, pp. 124-135 &copy; American Astronomical Society
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[[File:LedouxPekeris1941.jpg|600px|center|Ledoux &amp; Pekeris (1941, ApJ, 94, 124)]]
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If we divide their equation (1) through by <math>~Xr = \Gamma_1 P r</math> and recognize that,
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<math>
\frac{dX}{dr} = \frac{dX}{dm}\frac{dm}{dr} = - \Gamma_1 g_0 \rho \, ,
</math>
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we obtain,

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<math>
\frac{d^2\xi}{dr^2} + \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} +\frac{\rho}{\Gamma_1 P} \biggl[ \sigma^2 + (4 - 3\Gamma_1) \frac{g_0}{r} \biggr] \xi = 0 \, .
</math>
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Clearly, this 2<sup>nd</sup>-order, ordinary differential equation is the same as our derived LAWE, but with a more general definition of the adiabatic exponent that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.

Multiplying this last equation through by <math>~\Gamma_1 P r^4</math>, and recognizing that,

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<math>~(r^4 \Gamma_1 P)\frac{d^2\xi}{dr^2} </math>
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<math>~=</math>
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<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
</math>
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<span id="RewrittenLAWE">we can write,</span>
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<math>~0</math>
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<math>~=</math>
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<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \frac{d\xi}{dr} \cdot \frac{d}{dr} \biggl[ r^4 \Gamma_1 P\biggr]
+ ( \Gamma_1 P r^4 ) \biggl[ \frac{4}{r} - \frac{g_0 \rho}{P} \biggr] \frac{d\xi}{dr} 
+\rho \biggl[ \sigma^2 r^4 + (4 - 3\Gamma_1) g_0 r^3\biggr] \xi
</math>
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&nbsp;
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<math>~=</math>
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<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] - \biggl[4r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr}  \biggr] \frac{d\xi}{dr} 
+ \biggl[ 4 r^3\Gamma_1 P + \Gamma_1 r^4 \frac{dP}{dr}\biggr] \frac{d\xi}{dr} 
+\biggl[ \sigma^2 \rho r^4 - (4 - 3\Gamma_1) r^3 \frac{dP}{dr} \biggr] \xi
</math>
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(checked for n = 5) ==>
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<math>~=</math>
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<math>~
\frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] 
+\biggl[ \sigma^2 \rho r^4 + (3\Gamma_1 - 4) r^3 \frac{dP}{dr} \biggr] \xi 
</math>
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&nbsp;
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<math>~=</math>
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<math>~
\frac{d}{dr}\biggl[ \Gamma_1 P r^4 ~\frac{d\xi}{dr} \biggr] 
+\biggl[ \sigma^2 r^4 \rho  + 4 Gm (r ) r \rho + 3\Gamma_1 r^3 \frac{dP}{dr} \biggr] \xi \, .
</math>
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Assuming that <math>~\Gamma_1</math> is uniform throughout the configuration, this last expression is the same as equation (3) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux &amp; Pekeris (1941)], while the next-to-last expression is identical to equation (58.1) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux &amp; Walraven (1958)].