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Ledoux's Variational Principle (Supporting Derivations)
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An accompanying chapter presents a summary discussion of Ledoux's variational principle, emphasizing how it relates to stability analyses that are built upon freeenergy arguments. The detailed derivations presented below provide the foundation upon which that summary discussion has been built.
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Our Initial Explorations
Review by Ledoux and Walraven (1958)
Here we are especially interested in understanding the origin of equation (59.10) of P. Ledoux & Th. Walraven (1958), which appears in §59 (pp. 464  466) of their Handbuch der Physik article.
From our accompanying summary of the set of nonlinear governing relations, we highlight the
Euler + Poisson Equations
Repeating a result from our separate derivation, linearization of the two terms on the righthand side of this equation gives,






Adopting the terminology of Ledoux & Walraven (1958), the "variation" of each of these terms is obtained by subtracting off the leading order pieces — which presumably cancel in equilibrium. In particular, drawing a parallel with their equation (59.1), we can write,



And, drawing a parallel with their equation (59.2), we have,









Now, if we combined the linearized continuity equation and the linearized (adiabatic form of the) first law of thermodynamics, as derived elsewhere, we can write,






Hence,



So, given that a mapping from our notation to that used by Ledoux & Walraven (1958) requires , I understand the origins of their equations (59.1) and (59.2). But I do not yet understand how … "Accordingly, the acting forces per unit volume can be considered as deriving from a potential density"



It is clear that, once I understand the origin of this expression for the potential density, I will understand how the "Lagrangian density" as defined by their equation (47.8), viz.,
becomes (see their equation 59.5),



Noting that, , this in turn gives,


















The group of terms inside the curly braces, here, matches the group of terms inside the curly braces of Ledoux & Walraven's equation (59.8) if we acknowledge that:
 Our has the same meaning as, but the opposite sign of, their .
 Our last term goes to zero because, at the center, while at the surface.
LP41 Again
After setting the last term to zero, this last expression can be rewritten as,









As is explained in detail in §59 (pp. 464  465) of Ledoux & Walraven (1958), and summarized in §1 of Ledoux & Pekeris (1941), the function inside the curly braces of this last expression will be minimized if the radially dependent displacement function, , is set equal to the eigenfunction of the fundamental mode of radial oscillation, ; and, after evaluation, the minimum value of this expression will be equal to (the negative of) the square of the fundamentalmode oscillation frequency, . This explicit mathematical statement is contained within equation (8) of Ledoux & Pekeris and within equation (59.10) of Ledoux & Walraven.
Now, as we have discussed separately — see, also, p. 64, Equation (12) of [C67] — the gravitational potential energy of the unperturbed configuration is given by the integral,



for adiabatic systems, the internal energy is,
and — see the text at the top of p. 126 of Ledoux & Pekeris (1941) — the moment of inertia of the configuration about its center is,
Hence, the function to be minimized may be written as,



This expression appears in equation (9) of Ledoux & Pekeris (1941).
Chandrasekhar (1964)
In a paper titled, A General Variational Principle Governing the Radial and the NonRadial Oscillations of Gaseous Masses, S. Chandrasekhar (1964, ApJ, 139, 664) independently derived the LedouxPekeris Lagrangian.
The Lagrangian Expression using Chandrasekhar's Notation
First, let's show that the Lagrangian expression derived by Chandrasekhar is, indeed, equivalent to the one presented by Ledoux & Pekeris. Returning to the second line of our effort to simplify the above definition of the Lagrangian, and making the substitution, , we have,









This integral expression matches the integral expression that appears in equation (49) of Chandrasekhar (1964), if we accept that our squared frequency, , has the opposite sign to Chandrasekhar's . Chandrasekhar acknowledged that, for radial modes of oscillation, his result was the same as that derived earlier by Ledoux and his collaborators.
Chandrasekhar's Independent Derivation
Now, let's follow Chandrasekhar's lead and derive the Lagrangian directly from the governing LAWE. We begin with a version of the LAWE that appears above in our review of the paper by Ledoux & Pekeris (1941), namely,



We will develop the Lagrangian expression by following the guidance provided at the top of p. 666 of S. Chandrasekhar (1964, ApJ, 139, 664). First, we multiply the LAWE through by the fractional displacement, ; second, we make the substitution, , in order to shift to Chandrasekhar's variable notation; then we multiply through by and integrate from the center to the surface of the configuration.
Multiplying through by the fractional displacement gives,



Next, making the stated variable substitution gives,


















Finally, integrating over the volume gives,



which is identical to equation (49) of Chandrasekhar (1964), if the last term — the difference of the central and surface boundary conditions — is set to zero.
Examples
Ledoux's Expression
Returning to the last line of our above definition of the Lagrangian, that is,



let's attempt to evaluate the terms inside the curly braces for the case of pressuretruncated polytropic configurations because, as has been discussed separately, we have an analytic expression for the eigenvector of the fundamentalmode of radial oscillation. Dividing through by and making the substitution, , gives,












where, we have set the pressure at the (truncated) surface to the value, .
Chandra's Expression
Normalization
Alternatively, starting from Chandrasekhar's expression,















Known Analytic Eigenfunction
Now, let's plug in the known eigenfunction for the marginally unstable configuration, namely,
Exact Solution to the Polytropic LAWE  

and 

Given that, from the LaneEmden equation,
we recognize that,












General Evaluation
Therefore, returning to Chandrasekhar's expression for the Lagrangian and evaluating the sum of the last two terms inside the curly braces gives,







































Application to n = 5 Polytropic Configuration
Let's try plugging in expressions for n = 5 configurations, for which the LaneEmden function is known analytically. Specifically,
and
First Attempt
We have,

































Second Attempt
First, we evaluate,









and,









Then, returning to Chandrasekhar's expression for the Lagrangian and evaluating the sum of the last two terms inside the curly braces gives,


















Normalizations
Returning to the last line of our derivation of the Ledoux & Walraven Lagrangian, we can write,









Alternate Approach: Integrate Over LAWE
As we have demonstrated, above, if we assume that is constant throughout the configuration, our version of the LAWE can be straightforwardly rearranged to give equation (58.1) of Ledoux & Pekeris (1941), that is,



where, we are using in place of to represent the fractional Lagrangian displacement, . If we multiply this expression through by and integrate over the entire volume of the configuration, we have,












where the definitions of and are as provided, above. This last expression is the same as equation (59.17) of Ledoux & Pekeris (1941), except that: these authors have retained a term allowing for radial variation of , whereas we have not; and we have retained a boundary term that can accommodate a nonzero surface pressure, whereas Ledoux & Pekeris have not.
See Also
© 2014  2021 by Joel E. Tohline 