Editing SSC/PerspectiveReconciliation (section)

==Lagrangian Reformulation==

Quite generally, we can rewrite the ''Lagrangian-formulated'' wave equation as,
<div align="center">
<math>
\biggl(\frac{P_0}{P_c}\biggr)\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr) 
- \biggl(\frac{\rho_0}{\rho_c}\biggr)  \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} 
+ \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr]  x = 0 .
</math><br />
</div>
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, <math>~n=1</math> polytrope are plugged into it, as is demonstrated in an [[SSC/Stability/Polytropes#MurphyFiedler1985b|accompanying discussion]], we obtain exactly the same 2<sup>nd</sup>-order ODE as published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985)].  For an homogeneous sphere, in particular, this expression can be rewritten as follows.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 )\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4(1-\chi_0^2)}{\chi_0} 
-  2\chi_0 \biggr] \frac{dx}{d\chi_0} 
+ \frac{1}{\gamma_\mathrm{g}} \biggl[2\biggl(\frac{\omega^2 R^3}{GM}\biggr) + 2(4 - 3\gamma_\mathrm{g}) \biggr]  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 )\frac{d^2x}{d\chi_0^2} + \frac{1}{\chi_0}\biggl[4 -  6\chi_0^2 \biggr] \frac{dx}{d\chi_0} + \mathfrak{F}  x \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\mathfrak{F}  \equiv \frac{2}{\gamma_\mathrm{g}} \biggl[\biggl(\frac{\omega^2 R^3}{GM}\biggr) + (4 - 3\gamma_\mathrm{g}) \biggr]  \, .</math>
</div>
This expression precisely matches equation (2) of [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)].

Drawing from [[SSC/Stability/UniformDensity#Sterne.27s_Presentation|Sterne's presentation]], the following table details the eigenfunctions for the four lowest radial modes that satisfy this wave equation.

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr><th colspan="4">Sterne's (1937) Eigenfunctions for Homogeneous Sphere</th></tr>
<tr>
  <td align="center">Mode</td>
  <td align="center" rowspan="2">Eigenvector</td>
  <td align="center">Square of Eigenfrequency:<p></p><math>~3\omega^2/(4\pi G\rho)</math></td>
  <td align="center" rowspan="6">[[File:Sterne1937SolutionPlot1.png|350px|center|Sterne (1937)]]</td>
</tr>
<tr>
  <td align="center"><math>~j</math></td>
  <td align="center"><math>~\gamma[3+j(2j+5)] - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~0</math></td>
  <td align="left"><math>~x = 1</math></td>
  <td align="center"><math>~3\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="left"><math>~x = 1 -\frac{7}{5} \chi_0^2</math></td>
  <td align="center"><math>~10\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="left"><math>~x = 1 -\frac{18}{5} \chi_0^2 + \frac{99}{35} \chi_0^4</math></td>
  <td align="center"><math>~21\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="left"><math>~x = 1 -\frac{33}{5} \chi_0^2 + \frac{429}{35} \chi_0^4 - \frac{143}{21} \chi_0^6</math></td>
  <td align="center"><math>~36\gamma-4</math></td>
</tr>
</table>
</div>