Editing ThreeDimensionalConfigurations/EFE Energies (section)

===Conserve Only L===
Let's fix the total angular momentum, <math>~L</math>, of a triaxial configuration and examine how the configuration's free energy varies as we allow it to contort through different triaxial shapes &#8212; that is, as its pair of axis ratios varies, always maintaining <math>~\tfrac{b}{a} < 1</math> &#8212; and as we vary <math>~x</math>, which characterizes the fraction of angular momentum that is stored in internal spin versus overall figure rotation.  The desired free-energy function, <math>~E(\tfrac{b}{a},\tfrac{c}{a}, x)|_L</math>, has [[#E_Lexpression|just been defined]], but visualizing its behavior is difficult because, in this situation, the free energy is a warped, ''three-dimensional'' surface draped across the four-dimensional domain, <math>~(\tfrac{b}{a},\tfrac{c}{a}, x, E_L)</math>.   


Acknowledging that we are primarily interested in identifying extrema of this free-energy function, the discussion presented in &sect;3.2 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] shows us how to reduce the dimensionality of this problem by one.  There, it is shown that, as long as <math>~\tfrac{b}{a} \ne 1</math>, extrema exist in the <math>~x</math>-coordinate direction &#8212; that is, <math>~\partial E_L/\partial x = 0</math> &#8212; only if <math>~x = 0.</math>  For a given choice of <math>~L</math>, therefore, the relevant ''two-dimensional'' free-energy surface is defined by the expression,

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<math>~E\biggl(\frac{b}{a}, \frac{c}{a}, x=0\biggr)\biggr|_L</math>
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<math>~=</math>
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<math>~\frac{L^2}{2} \biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1}  \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{-2/3} 
\biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2\biggr]^{-1} 
</math>
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&nbsp;
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&nbsp;
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<math>~
- 2\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1}  \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} 
\biggl[A_1 + A_2\biggl(\frac{b}{a}\biggr)^2 + A_3\biggl(\frac{c}{a}\biggr)^2 \biggr]\, .</math>
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Figure 3 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] presents a black-and-white contour plot of this <math>~E_L</math> function for the specific case of <math>~L = 4.71488</math>, which, for reference, is the total angular momentum of an equilibrium [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Maclaurin spheroid]] having an eccentricity, <math>~e = 0.85</math> (see [[#Table1|Table 1, below]]).  We have digitally extracted this black-and-white contour plot from p. 477 of the (PDF-formatted) [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] publication and have reprinted it as the left-hand panel of our Figure 1.  Note that we have flipped the plot horizontally and rotated it by 90&deg; so that the orientation of the axis pair, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math>, conforms with the orientation of a related, information-rich diagram presented by [https://ui.adsabs.harvard.edu/abs/1965ApJ...142..890C/abstract Chandrasekhar (1965)] &#8212; see also our [[ThreeDimensionalConfigurations/JacobiEllipsoids#Sequence_Plots|accompanying discussion of equilibrium sequence plots]].  

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<th align="center" colspan="1"><font size="+1">Figure 1:</font>  Free-Energy Surface Projected onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> Plane
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[[File:JacobiPaperIFig3flipped.png|240px|Christodoulou1995Fig3 Flipped]]
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[[File:VisTrailsFig3f.png|240px|Both 2D contour plots overlaid]]
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[[File:VisTrailsFig3d.png|240px|Our 2D colored contour plot]]
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All three contour plots show how the free-energy, <math>~E_L</math>, varies across the <math>~(\tfrac{b}{a}, \tfrac{c}{a})</math> domain for the specific case of <math>~L = 4.71488</math>.  Horizontal axis is <math>~0 \le \tfrac{b}{a} \le 1</math> and vertical axis is  <math>~0 \le \tfrac{c}{a} \le 1</math>.
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  <td align="center" width="240px"><b>Left-hand Panel:</b><br />Black-and-white contour plot<br />
extracted from p. 477 of [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I]<br />
"''Phase-Transition Theory of Instabilities.  I. Second-Harmonic Instability and Bifurcation Points''"<p></p>
ApJ, vol. 446, pp. 472-484 &copy; [http://aas.org/ AAS]
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  <td align="center" width="240px"><b>Middle Panel:</b><br />Black-and-white contour plot digitally overlaid on color contour plot.</td>
  <td align="center" width="240px"><b>Right-hand Panel:</b><br />Color contour plot<br />created here as a projection of the free-energy surface shown in Fig. 2.</td>
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In our Figure 2, this same <math>~E_L</math> function has been displayed as a warped, two-dimensional free-energy surface draped across the three-dimensional <math>~(\tfrac{b}{a},\tfrac{c}{a},E)</math> domain, where depth as well as color has been used to tag energy values.  The two-dimensional, colored contour plot presented in the right-hand panel of our Figure 1 results from the projection of this free-energy surface onto the <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> plane; it reproduces in quantitative detail the black-and-white contour plot that we have extracted from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I].  In an effort to (qualitatively) illustrate this agreement, we have digitally "pasted" the black-and-white contour plot from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I] onto our colored contour plot and presented the combined image in the middle panel of our Figure 1. 


Our Figure 2 image of the free-energy surface helps illuminate the description of this surface that appears in the caption of Fig. 3 from [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I].  Quoting from that figure caption:&nbsp; "The [equilibrium] Maclaurin spheroid sits on a saddle point <math>~[(\tfrac{b}{a},\tfrac{c}{a}) = (1.0,0.52678); E_0 = -7.81842]</math>, while a global minimum with  <math>~E_0 = -7.83300</math> exists at <math>~(\tfrac{b}{a},\tfrac{c}{a}) = (0.588,0.428)</math>."

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<th align="center" colspan="1"><font size="+1">Figure 2:</font>  Free-Energy Surface
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[[File:VistrailsFig3b.png|600px|Christodoulou1995Fig3 Flipped]]
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