User:Tohline/SSC/Stability/MoreGeneralApproach

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More General Approach to the Parabolic Eigenvalue Problem

Whitworth's (1981) Isothermal Free-Energy Surface
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The material presented in this chapter is an extension of the chapter titled, Other Analytic Models and could also be considered to be a subsection of the associated chapter titled, Other Analytic Ramblings. More specifically, in the following "Introduction," we repeat a manipulation of the LAWE that was originally developed in the subsection of that chapter titled, "Consider Parabolic Case".



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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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Introduction

In the case of a parabolic density distribution, the LAWE may be written in the form,

<math>~\frac{2}{(1-x^2)(2-x^2)} \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>

<math>~=</math>

<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>


Let's try,

<math>~\mathcal{G}_\sigma</math>

<math>~=</math>

<math>~(a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^m \, ,</math>

which implies,

<math>~\mathcal{G}_\sigma^'</math>

<math>~=</math>

<math>~n(a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^m +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2x)</math>

<math>~\Rightarrow ~~~~ \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}</math>

<math>~=</math>

<math>~n(a_0 + a_2x^2)^{-1}(2a_2x^2) +m (b_0 + b_2x^2)^{-1}(2b_2x^2) </math>

 

<math>~=</math>

<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ n a_2 (b_0 + b_2x^2) +mb_2 (a_0 + a_2x^2) \biggr] </math>

 

<math>~=</math>

<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0) +(na_2 b_2+ mb_2 a_2)x^2\biggr] \, ,</math>

and,

<math>~\mathcal{G}_\sigma^{' '}</math>

<math>~=</math>

<math>~n m (a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) + n(a_0 + a_2x^2)^{n-1}(2a_2) \cdot (b_0 + b_2x^2)^m + n(n-1)(a_0 + a_2x^2)^{n-2}(2a_2x)^2 \cdot (b_0 + b_2x^2)^m </math>

 

 

<math>~+m n(a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2) +m(m-1) (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-2}(2b_2x)^2</math>

<math>~\Rightarrow ~~~~ \frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}</math>

<math>~=</math>

<math>~8n m a_2b_2 x^2 (a_0 + a_2x^2)^{-1}\cdot (b_0 + b_2x^2)^{-1} + n2a_2 (a_0 + a_2x^2)^{-1} + n(n-1)4a_2^2 x^2 (a_0 + a_2x^2)^{-2} +m2b_2 (b_0 + b_2x^2)^{-1} +m(m-1)4 b_2^2 x^2 (b_0 + b_2x^2)^{-2}</math>

 

<math>~=</math>

<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2 (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)} + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}} + \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2 </math>

So, we have for the LAWE:

LHS

<math>~=</math>

<math>~ \frac{2}{(1-x^2)(2-x^2)} \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>

 

<math>~=</math>

<math>~ \frac{2}{(1-x^2)(2-x^2)(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl\{ \biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0) + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) </math>

 

 

<math>~ -\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) \biggr\} \, ;</math>

RHS

<math>~=</math>

<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>

 

<math>~=</math>

<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2 (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)} + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}} + \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2 </math>

 

 

<math>~ + \frac{8}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0) +(na_2 b_2+ mb_2 a_2)x^2\biggr] </math>

 

<math>~=</math>

<math>~\frac{1}{(a_0 + a_2x^2)(b_0 + b_2x^2)} \biggl\{ 2n a_2(b_0 + b_2x^2) + 2m b_2 (a_0 + a_2x^2) + 8(n a_2 b_0 + mb_2 a_0) + 8(na_2 b_2+ mb_2 a_2)x^2 </math>

 

 

<math>~ + \biggl[8n m a_2b_2+ \frac{4n(n-1) a_2^2(b_0 + b_2x^2) }{ (a_0 + a_2x^2)} + \frac{4m(m-1) b_2^2(a_0 + a_2x^2) }{(b_0 + b_2x^2)} \biggr]x^2 \biggr\} \, . </math>

Putting these together gives,

<math>~ 0 </math>

<math>~=</math>

<math>~ \biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0) + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) -\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) </math>

 

 

<math>~ - \biggl[ n a_2(b_0 + b_2x^2) + m b_2 (a_0 + a_2x^2) + 4(n a_2 b_0 + mb_2 a_0) + 4(na_2 b_2+ mb_2 a_2)x^2+ 4n m a_2b_2x^2 \biggr](1-x^2)(2-x^2) </math>

 

 

<math>~ - \frac{(1-x^2)(2-x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}\biggl[2n(n-1) a_2^2(b_0 + b_2x^2)^2 + 2m(m-1) b_2^2(a_0 + a_2x^2)^2 \biggr]x^2 \, . </math>

Additional Setup

Benefitting from our earlier exploration of this problem, let's divide through by the product, <math>~(a_0 b_0)</math>, and introduce the new variable notations,

<math>~\lambda \equiv \frac{a_2}{a_0} \, ,</math>       and       <math>~\eta \equiv \frac{b_2}{b_0} \, .</math>

The LAWE becomes,

<math>~ 0 </math>

<math>~=</math>

<math>~ \biggl[ \alpha(1 + \lambda x^2) (1 + \eta x^2) + 2x^2(n \lambda + m\eta ) + 2x^4 (n\lambda \eta + m\eta \lambda ) \biggr](5-3x^2) -\sigma^2 (1 + \lambda x^2) (1 + \eta x^2) </math>

 

 

<math>~ - \biggl[ n \lambda (1 + \eta x^2) + m \eta (1 + \lambda x^2) + 4(n \lambda + m\eta ) + 4(n\lambda \eta + m\eta \lambda )x^2+ 4n m \lambda \eta x^2 \biggr](1-x^2)(2-x^2) </math>

 

 

<math>~ - \frac{(1-x^2)(2-x^2)}{ (1 + \lambda x^2)(1 + \eta x^2)}\biggl[2n(n-1) \lambda^2(1 + \eta x^2)^2 + 2m(m-1) \eta ^2(1 + \lambda x^2)^2 \biggr]x^2 \, . </math>



Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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