Editing Appendix/Ramblings/LedouxVariationalPrinciple (section)

==Ledoux's Expression==
Returning to the last line of our [[#LDefinition|above definition of the Lagrangian]], that is,

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<math>~L </math>
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<math>~=</math>
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<math>~ 2\pi e^{2i\omega t} \biggl\{ 
- \int_0^R \rho_0 \omega^2 r_0^4 x^2 dr_0
- \int_0^R  \gamma_\mathrm{g} P_0 r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2  dr_0
+ \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)P_0\biggr]dr_0
-\biggl[3 \gamma_\mathrm{g} r_0^3 x^2 P_0\biggr]_0^{R}
\biggr\} \, ,
</math>
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let's attempt to evaluate the terms inside the curly braces for the case of pressure-truncated polytropic configurations because, as has been discussed separately, we have an analytic expression for the eigenvector of the fundamental-mode of radial oscillation.  Dividing through by <math>~P_c R_\mathrm{eq}^3</math> and making the substitution, <math>~r_0/R_\mathrm{eq} \rightarrow \xi/\tilde\xi</math>, gives,

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<math>~\frac{L_{\{\}} }{P_c R_\mathrm{eq}^3}</math>
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<math>~=</math>
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<math>~
- \int_0^R \frac{\rho_0 \omega^2}{P_c R_\mathrm{eq}^3} r_0^4 x^2 dr_0
- \int_0^R  \gamma_\mathrm{g} \frac{P_0}{P_c R_\mathrm{eq}^3} r_0^4\biggl( \frac{\partial x}{\partial r_0}\biggr)^2  dr_0
+ \int_0^R r_0^3 x^2 \frac{d}{dr_0}\biggl[ (3\gamma_\mathrm{g} - 4)\frac{P_0}{P_c R_\mathrm{eq}^3}\biggr]dr_0
- 3 \gamma_\mathrm{g} x_\mathrm{surf}^2 \frac{P_e}{P_c}
</math>
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&nbsp;
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<math>~=</math>
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<math>~
- \int_0^{\tilde\xi} \omega^2 \biggl[\frac{\rho_c  R_\mathrm{eq}^2}{P_c } \biggr] \biggl( \frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^4 x^2 \frac{d\xi}{\tilde\xi}
- \int_0^{\tilde\xi}  \gamma_\mathrm{g} \biggl(\frac{P_0}{P_c }\biggr) \biggl(\frac{\xi}{\tilde\xi}\biggr)^4\biggl[ \frac{\partial x}{\partial (\xi/\tilde\xi)}\biggr]^2  \frac{d\xi}{\tilde\xi}
+ \int_0^{\tilde\xi} \biggl(\frac{\xi}{\tilde\xi}\biggr) x^2 \frac{d}{d(\xi/\tilde\xi)}\biggl[ (3\gamma_\mathrm{g} - 4)\frac{P_0}{P_c }\biggr] \frac{d\xi}{\tilde\xi}
- 3 \gamma_\mathrm{g} x_\mathrm{surf}^2 \frac{P_e}{P_c}
</math>
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&nbsp;
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<math>~=</math>
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<math>~
- \omega^2 \biggl[\frac{\rho_c  R_\mathrm{eq}^2}{P_c ~{\tilde\xi}^5} \biggr]  \int_0^{\tilde\xi} \theta^n \xi^4 x^2 d\xi
- \frac{\gamma_\mathrm{g}}{ {\tilde\xi}^3} \int_0^{\tilde\xi}   \theta^{n+1} \xi^4\biggl[ \frac{\partial x}{\partial \xi}\biggr]^2  d\xi
+ \frac{(3\gamma_\mathrm{g} - 4)}{\tilde\xi}\int_0^{\tilde\xi} \xi x^2 \frac{d}{d\xi}\biggl[ \theta^{n+1}\biggr] d\xi
- \biggl[\frac{3 \gamma_\mathrm{g}P_e}{P_c} \biggr]x_\mathrm{surf}^2 
</math>
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&nbsp;
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<math>~=</math>
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<math>~
- \omega^2 \biggl[\frac{\rho_c  R_\mathrm{eq}^2}{P_c ~{\tilde\xi}^5} \biggr]  \int_0^{\tilde\xi} \theta^n \xi^4 x^2 d\xi
- \biggl[\frac{3 \gamma_\mathrm{g}P_e}{P_c} \biggr]x_\mathrm{surf}^2 
+  \frac{1}{ {\tilde\xi}^3} \int_0^{\tilde\xi} \biggl[ (3\gamma_\mathrm{g} - 4) {\tilde\xi}^2 \xi x^2 \frac{d\theta^{n+1}}{d\xi}
- \gamma_\mathrm{g} \theta^{n+1} \xi^4\biggl( \frac{\partial x}{\partial \xi}\biggr)^2  \biggr]d\xi
</math>
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where, we have set the pressure at the (truncated) surface to the value, <math>~P_0|_\mathrm{surface} = P_e</math>.