Editing Appendix/Ramblings/LedouxVariationalPrinciple (section)

==LP41 Again==
After setting the last term to zero, this last expression can be rewritten as,

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<math>~2 e^{-2i\omega t} L </math>
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<math>~=</math>
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<math>~ 
- \omega^2 \int_0^R 4\pi\rho_0 r_0^4 x^2 dr_0
- \int_0^R  4\pi \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2  dr_0
- (3\gamma_\mathrm{g} - 4)\int_0^R 4\pi\rho_0 r_0^3 x^2 \biggl( -\frac{1}{\rho_0}\frac{dP_0}{dr_0} \biggr)dr_0
</math>
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&nbsp;
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<math>~=</math>
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<math>~ 
- \omega^2 \int_0^R x^2 r_0^2 dm
- \gamma_\mathrm{g} \int_0^R  \biggl[ r_0 \biggl( \frac{\partial x}{\partial r_0}\biggr) \biggr]^2 P_0  dV
- (3\gamma_\mathrm{g} - 4) \int_0^R  r_0 x^2 \biggl( \frac{Gm}{r_0^2} \biggr)dm
</math>
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<math>~- \biggl[ \frac{2 e^{-2i\omega t}}{\int_0^R x^2 r_0^2 dm } \biggr] L </math>
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<math>~=</math>
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<math>~ 
\omega^2 
+ \biggl\{ \frac{\gamma_\mathrm{g} \int_0^R  \bigl[ r_0 \bigl( \frac{\partial x}{\partial r_0}\bigr) \bigr]^2 P_0  dV
+ (3\gamma_\mathrm{g} - 4) \int_0^R  x^2 \bigl( \frac{Gm}{r_0} \bigr)dm}{\int_0^R x^2 r_0^2 dm}
\biggr\} \, .
</math>
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As is explained in detail in &sect;59 (pp. 464 - 465) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux &amp; Walraven (1958)], and summarized in &sect;1 of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux &amp; Pekeris (1941)], the function inside the curly braces of this last expression will be minimized if the radially dependent displacement function, <math>~x</math>, is set equal to the eigenfunction of the fundamental mode of radial oscillation, <math>~x_0</math>; and, after evaluation, the minimum value of this expression will be equal to (the negative of) the square of the fundamental-mode oscillation frequency, <math>~\omega^2</math>.  This explicit mathematical statement is contained within equation (8) of Ledoux &amp; Pekeris and within equation (59.10) of Ledoux &amp; Walraven.

<span id="EnergiesDefined">Now,</span> 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_g}-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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Hence, the function to be minimized may be written as,

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&nbsp;
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&nbsp;
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<math>~ 
\biggl\{ \frac{\gamma_\mathrm{g} (\gamma_\mathrm{g}-1) \int_0^R  \bigl[ r_0 \bigl( \frac{\partial x}{\partial r_0}\bigr) \bigr]^2 dU_\mathrm{int}
- (3\gamma_\mathrm{g} - 4) \int_0^R  x^2 dW_\mathrm{grav}}{\int_0^R x^2 dI}
\biggr\} \, .
</math>
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This expression appears in equation (9) of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux &amp; Pekeris (1941)].