Editing SSC/VariationalPrinciple (section)

===Ledoux &amp; Walraven Approach===
Returning to the above [[#FoundationalVariationalRelation|''Foundational Variational Relation'']], we can also write,

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<math>~\sigma^2 \rho r^4 \xi^2</math>
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<math>~=</math>
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<math>~
-\xi \cdot \frac{d}{dr}\biggl[ r^4 \Gamma_1 P ~\frac{d\xi}{dr} \biggr] 
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) 
</math>
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&nbsp;
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<math>~=</math>
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<math>~
r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2  
- (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) 
- \frac{d}{dr}\biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]
</math>
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<math>~\Rightarrow ~~~ \int_0^R\sigma^2 \rho r^4 \xi^2 dr</math>
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<math>~=</math>
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<math>~
\int_0^R r^4 \Gamma_1 P \biggl(\frac{d\xi}{dr}\biggr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \biggl( \frac{dP}{dr} \biggr) dr
- \biggr[r^4 \Gamma_1 P\xi \biggl(\frac{d\xi}{dr}\biggr) \biggr]_0^R
</math>
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If the last term (boundary conditions) is set to zero, then we may also write,
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<math>~\sigma^2 </math>
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<math>~=</math>
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<math>~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
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This means that, if the radial profile of the pressure and the density is known throughout a spherically symmetric, equilibrium configuration, and if, furthermore, the eigenfunction, <math>~\xi(r)</math>, of a radial oscillation mode is specified precisely, then this expression will give the (square of the) ''eigenfrequency'' of that oscillation mode.

By using formal ''variational principle'' techniques to derive this same expression, [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux &amp; Walraven (1958)] are able to offer a broader interpretation, which is encapsulated by their equation (59.10), viz.,
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<math>~\sigma_0^2 </math>
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<math>~=</math>
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<math>~\mathrm{min}~
\frac{\int_0^R r^4 \Gamma_1 P \bigl(\frac{d\xi}{dr}\bigr)^2 dr
- \int_0^R (3\Gamma_1 - 4) r^3 \xi^2 \bigl( \frac{dP}{dr} \bigr) dr}{\int_0^R \rho r^4 \xi^2 dr} \, .
</math>
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This means that, if the exact radial eigenfunction, <math>~\xi(r)</math>, is not known, various approximate eigenfunctions can be tried.  The trial eigenfunction that ''minimizes'' the righthand-side of this expression will give the (square of the) eigenfrequency of the ''fundamental'' mode of oscillation (subscript zero).  Furthermore, via an evaluation of this righthand-side expression, any reasonable trial eigenfunction &#8212; for example, <math>~\xi</math> = constant &#8212; can provide an ''upper limit'' to  <math>~\sigma_0^2</math>.