Appendix/Ramblings/LedouxVariationalPrinciple - Revision history

Joel2: /* Application to n = 5 Polytropic Configuration */

2024-07-05T23:46:03Z

Application to n = 5 Polytropic Configuration

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	===Application to n = 5 Polytropic Configuration===		===Application to n = 5 Polytropic Configuration===

	Let's try plugging in expressions for n = 5 configurations, for which the [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29\|Lane-Emden function is known analytically]]. Specifically,		Let's try plugging in expressions for n = 5 configurations, for which the [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29\|Lane-Emden function is known analytically]]. Specifically,
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	<math>~\theta_{n=5} = \biggl(1 + \frac{\xi^2}{3}\biggr)^{-1 / 2}</math>       and       <math>~\theta_{n=5}^' = - \frac{\xi}{3} \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2} \, .</math>		<math>~\theta_{n=5} = \biggl(1 + \frac{\xi^2}{3}\biggr)^{-1 / 2}</math>       and       <math>~\theta_{n=5}^' = - \frac{\xi}{3} \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2} \, .</math>

Joel2: /* Known Analytic Eigenfunction */

2024-07-05T23:45:35Z

Known Analytic Eigenfunction

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	===Known Analytic Eigenfunction===		===Known Analytic Eigenfunction===
	Now, let's plug in the [[~~User:Tohline/~~SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations\|known eigenfunction for the marginally unstable configuration]], namely,		Now, let's plug in the [[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations\|known eigenfunction for the marginally unstable configuration]], namely,

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Joel2: /* LP41 Again */

2024-07-05T23:44:44Z

LP41 Again

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	As is explained in detail in §59 (pp. 464 - 465) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)], and summarized in §1 of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & 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 & Pekeris and within equation (59.10) of Ledoux & Walraven.		As is explained in detail in §59 (pp. 464 - 465) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)], and summarized in §1 of [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & 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 & Pekeris and within equation (59.10) of Ledoux & Walraven.

	<span id="EnergiesDefined">Now,</span> as we have [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Wgrav\|discussed separately]] — see, also, p. 64, Equation (12) of [<b>[[~~User:Tohline/~~Appendix/References#C67\|<font color="red">C67</font>]]</b>] — the gravitational potential energy of the unperturbed configuration is given by the integral,		<span id="EnergiesDefined">Now,</span> as we have [[SSCpt1/Virial#Wgrav\|discussed separately]] — see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>] — the gravitational potential energy of the unperturbed configuration is given by the integral,
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	for adiabatic systems, the [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Reservoir\|internal energy]] is,		for adiabatic systems, the [[SSCpt1/Virial#Reservoir\|internal energy]] is,
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	<math>		<math>

Joel2: /* Review by Ledoux and Walraven (1958) */

2024-07-05T23:38:44Z

Review by Ledoux and Walraven (1958)

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	Here we are especially interested in understanding the origin of equation (59.10) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)], which appears in §59 (pp. 464 - 466) of their ''Handbuch der Physik'' article.		Here we are especially interested in understanding the origin of equation (59.10) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)], which appears in §59 (pp. 464 - 466) of their ''Handbuch der Physik'' article.

	From our [[~~User:Tohline/~~SSC/Perturbations#Summary_Set_of_Nonlinear_Governing_Relations\|accompanying summary of the set of nonlinear governing relations]], we highlight the		From our [[SSC/Perturbations#Summary_Set_of_Nonlinear_Governing_Relations\|accompanying summary of the set of nonlinear governing relations]], we highlight the

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	Repeating a result from our [[~~User:Tohline/~~SSC/Perturbations#Euler_.2B_Poisson_Equations\|separate derivation]], linearization of the two terms on the righthand side of this equation gives,		Repeating a result from our [[SSC/Perturbations#Euler_.2B_Poisson_Equations\|separate derivation]], linearization of the two terms on the righthand side of this equation gives,

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	Now, if we combined the linearized continuity equation and the linearized (adiabatic form of the) first law of thermodynamics, as [[~~User:Tohline/~~SSC/Perturbations#Summary_Set_of_Linearized_Equations\|derived elsewhere]], we can write,		Now, if we combined the linearized continuity equation and the linearized (adiabatic form of the) first law of thermodynamics, as [[SSC/Perturbations#Summary_Set_of_Linearized_Equations\|derived elsewhere]], we can write,
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Joel2: /* Ledoux's Variational Principle (Supporting Derivations) */

2024-07-05T23:37:15Z

Ledoux's Variational Principle (Supporting Derivations)

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	=Ledoux's Variational Principle (Supporting Derivations)=		=Ledoux's Variational Principle (Supporting Derivations)=

	An [[~~User:Tohline/~~SSC/VariationalPrinciple#Ledoux.27s_Variational_Principle\|accompanying chapter]] presents a summary discussion of Ledoux's variational principle, emphasizing how it relates to stability analyses that are built upon free-energy arguments. The detailed derivations presented below provide the foundation upon which that summary discussion has been built.		An [[SSC/VariationalPrinciple#Ledoux.27s_Variational_Principle\|accompanying chapter]] presents a summary discussion of Ledoux's variational principle, emphasizing how it relates to stability analyses that are built upon free-energy arguments. The detailed derivations presented below provide the foundation upon which that summary discussion has been built.

	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

Joel2: Created page with "FORCETOC =Ledoux's Variational Principle (Supporting Derivations)= An accompanying chapter presents a summary discussion of Ledoux's variational principle, emphasizing how it relates to stability analyses that are built upon free-energy arguments. The detailed derivations presented below provide the foundation upon which that summary discussion has bee..."

2024-07-05T23:36:52Z

Created page with "__FORCETOC__  =Ledoux's Variational Principle (Supporting Derivations)= An accompanying chapter presents a summary discussion of Ledoux's variational principle, emphasizing how it relates to stability analyses that are built upon free-energy arguments. The detailed derivations presented below provide the foundation upon which that summary discussion has bee..."

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