Editing SSC/VariationalPrinciple (section)

===Ledoux &amp; Pekeris Approach===
Here we follow the lead of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux &amp; Pekeris (1941)].  Returning to the integral expression just derived in our discussion of the ''Ledoux &amp; Walraven approach'', and multiplying through by <math>~4\pi</math>, we have,

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<math>~\int_0^R 4\pi \sigma^2 \rho r^4 \xi^2 dr</math>
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<math>~=</math>
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<math>~
\int_0^R 4\pi r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) 4\pi r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[4\pi r^3 \Gamma_1 P\xi^2 \biggl(\frac{d\ln \xi}{d\ln r}\biggr) \biggr]_0^R \, .
</math>
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If we acknowledge that: 
* at the center of the configuration, <math>~r^3 = 0</math>; 
* [[#SurfaceBC|as above]], the boundary condition at the surface is <math>~P = P_e</math> while <math>~(d\ln \xi/d\ln r) = -3</math>; 
* the differential mass element is, <math>~dm = 4\pi r^2 \rho dr</math> and the corresponding differential volume element is, <math>~dV = 4\pi r^2 dr</math>; and
* a statement of detailed force balance is, <math>~dP/dr = - Gm\rho/r^2</math>,
this integral relation becomes,

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<math>~ \sigma^2 \int_0^R r^2 \xi^2 dm</math>
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<math>~=</math>
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<math>~
\Gamma_1 \int_0^R \biggl[ r \biggl(\frac{d\xi}{dr}\biggr)\biggr]^2 P dV
+ (3\Gamma_1 - 4) \int_0^R  \xi^2 \biggl( \frac{Gm}{r} \biggr) dm
- \biggr[\Gamma_1 \xi_\mathrm{surface}^2 (3P_e V) \biggl(-3\biggr) \biggr] \, .
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Now, as we have [[SSCpt1/Virial#Wgrav|discussed separately]] &#8212; see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] &#8212; the gravitational potential energy of the unperturbed configuration is given by the integral,
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<math>~W_\mathrm{grav}</math>
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<math>~=</math>
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<math>~ - \int_0^{M} \biggl( \frac{Gm}{r_0} \biggr) dm  \, ;</math>
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for adiabatic systems, the [[SSCpt1/Virial#Reservoir|internal energy]] is,
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<math>
U_\mathrm{int} 
=  \frac{1}{(\Gamma_1-1)} \int_0^R  P_0 dV
 \, ;</math>
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and &#8212; see the text at the top of p. 126 of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux &amp; Pekeris (1941)] &#8212; the moment of inertia of the configuration about its center is,
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<math>
I =   \int_0^M r_0^2 dm
 \, .</math>
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(Note that, defined in this way, <math>~I</math> is the same as [[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 [[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>

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  <td align="center" colspan="3"><font color="maroon"><b>Variational Principle's Governing Integral Relation</b></font></td>
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<math>~ \sigma^2 \int_0^R \xi^2 dI</math>
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<math>~=</math>
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<math>~
\Gamma_1 (\Gamma_1 - 1) \int_0^R \xi^2 \biggl[ \frac{d\ln\xi}{d\ln r}\biggr]^2 dU_\mathrm{int}
- (3\Gamma_1 - 4) \int_0^R  \xi^2 dW_\mathrm{grav}
+ 3^2 \Gamma_1  P_e V \xi_\mathrm{surface}^2 \, .
</math>
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