View source for SSC/PerspectiveReconciliation

__FORCETOC__
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=Reconciling Eulerian versus Lagrangian Perspectives=

<table border="1" cellpadding="8" align="center" width="90%">
<tr>
  <th align="center" colspan="3">Adopting the Perturbation Notation<sup>&dagger;</sup> of [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)]</th>
</tr>
<tr>
  <th align="center">''Eulerian''</th><th align="center">''Lagrangian''</th>
</tr>
<tr>
<td align="center" width="50%">
  
<!-- EULERIAN-->
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>~\hat{e}_r r</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [r_0 + \delta r(r) e^{i\sigma t}] </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\hat{e}_r v</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [0 + v_r^'(r) e^{i\sigma t}] = \hat{e}_r[i\sigma \delta r(r) e^{i\sigma t} ]</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\rho_0 + \rho^'(r) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~P_0 + P^'(r) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\Phi_0 + \Phi^'(r) e^{i\sigma t} </math>
  </td>
</tr>
</table>

</td>
<td align="center" width="50%">
  
<!-- LAGRANGIAN-->
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>~\hat{e}_r r</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [r_0 + \delta r(m)  e^{i\sigma t}] </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\hat{e}_r v</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [0 + \delta v_r(m)  e^{i\sigma t}] = \hat{e}_r[ i\sigma \delta r(m)  e^{i\sigma t}]</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\rho_0 + \delta \rho(m) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~P_0 + \delta P(m) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\Phi_0 + \delta \Phi(m) e^{i\sigma t} </math>
  </td>
</tr>
</table>

</td>
</tr>
<tr>
  <td align="left" colspan="2"><sup>&dagger;</sup>See especially the following clarifying comments from, respectively, &sect;43 (p. 432) and &sect;55 (p. 452) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux &amp; Walraven (1958)]: 
* The time derivatives of Eulerian coordinates have no meaning; hence, the (components of) <math>~\vec{r}</math> must be treated as Lagrangian coordinates in the definition,

<div align="center"><math>~\frac{d\vec{r}}{dt} = \vec{v}(\vec{r},t) \, .</math></div>

* Consistent with this understanding, we shall always denote a ''perturbation'' of <math>~\mathbf{r}</math> by <math>~\delta\mathbf{r}</math> but for all dependent variables <math>~\psi</math> we shall distinguish between the local or Eulerian perturbations <math>~\psi^'</math> and the Lagrangian perturbations <math>~\delta\psi</math> taken in following the motion.
</td>
</tr>
</table>


==Introduction==
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>(poor attempt at)<br />Reconciliation</b>]]</font>
|}
===Eulerian Approach===
In a [[SSC/StabilityEulerianPerspective#Stability_of_Spherically_Symmetric_Configurations_.28Eulerian_Perspective.29|related discussion]], we have shown that, from a standard Eulerian perspective, the perturbation and linearization of the [[PGE#Principal_Governing_Equations|principal governing equations]] leads to an equation of continuity (EOC), equation of motion (EOM), and Poisson equation of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \nabla\cdot \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial}{\partial t}\biggl( \frac{\rho_1}{\rho_0} \biggr) + \vec{v} \cdot \frac{\nabla\rho_0}{\rho_0} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \vec{v}}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \nabla\Phi_1 - \nabla\biggl[ \frac{\rho_1}{\rho_0}\biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2\Phi_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G \rho_1 \, .
</math>
  </td>
</tr>
</table>
</div>
The standard <font color="red">''Eulerian''</font> approach to combining these expressions is to <font color="red">take the time-derivative of the EOC, take the divergence of the EOM (and combine with the Poisson equation), then add the two</font> to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial^2}{\partial t^2}\biggl( \frac{\rho_1}{\rho_0} \biggr) + \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial \vec{v}}{\partial t} 
- 4\pi G \rho_1 - \nabla^2 \biggl[ \frac{\rho_1}{\rho_0}\biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .
</math>
  </td>
</tr>
</table>
</div>

After defining the ''fractional Eulerian density variation'' as,
<div align="center">
<math>~s \equiv \frac{\rho_1}{\rho_0} \, ,</math>
</div>
this becomes what we have [[SSC/StabilityEulerianPerspective#EulerianWaveEquation|referred to elsewhere as the Eulerian Wave Equation]].

===Lagrangian Approach===
In [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|another related discussion]], we have shown that, from a standard Lagrangian perspective, the perturbation and linearization of the principal governing equations leads to an equation of continuity (EOC), and equation of motion (EOM) ''already combined'' with the Poisson equation of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 3x - r_0 \biggl(\frac{dx}{dr_0}\biggr) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\gamma P_0}{\rho_0} \biggl[ \frac{d}{dr_0}\biggl(d \biggr) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\omega^2 r_0 x + g_0\biggl[ 4x + \gamma d \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
The standard <font color="red">''Lagranian''</font> approach to combining these expressions is to <font color="red">take the radial-derivative of the EOC, then substitute the EOC's expression for <math>~d</math> as well as its radial derivative into the EOM</font> to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{\gamma P_0}{\rho_0} \biggl[ 4\biggl(\frac{dx}{dr_0}\biggr) + r_0 \frac{d^2x}{dr_0^2} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\omega^2 r_0 x + g_0\biggl\{ 4x - \gamma \biggl[ 3x + r_0 \biggl(\frac{dx}{dr_0}\biggr) \biggr] \biggr\} \, ,</math>
  </td>
</tr>
</table>
</div>
which, after rearrangement of terms, gives what we have [[SSC/Perturbations#2ndOrderODE|referred to elsewhere as the Adiabatic Wave (or Radial Pulsation) Equation]].

===Rosseland's Approach===
[[SSC/PerspectiveReconciliation#Rosseland.27s_Derivation_of_the_Wave_Equation|Rosseland's approach]] is to <font color="red">take the total time-derivative of the EOM &#8212; even before linearizing any of the governing equations &#8212; then combine it strategically with the EOC and (integrated) Poisson equation</font> to obtain,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] 
+ \biggl\{ - \frac{d}{dt}\biggl( \frac{\partial\Phi}{\partial r}\biggr) + \biggl( \frac{\partial \Phi}{\partial r}\biggr) \frac{2v_r}{r} \biggr\}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \gamma_g P \nabla\cdot \vec{v}\biggr] + \frac{4gv_r}{r}  
+ \cancelto{\mathrm{small}}{\frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr]} \, .
</math>
  </td>
</tr>
</table>
</div>

By recognizing that <math>~\vec{v} = 0</math> in the initial equilibrium state and, therefore, that after perturbing the system, <math>~v_r</math> is a "small quantity," this equation is already linearized because every term contains one factor of <math>~v_r</math>.  Actually, as indicated, the last term on the right-hand-side can be dropped because it is of order <math>~(v_r)^2</math>.


===Approach by Ledoux and Walraven===
Note: It also might prove valuable to look at the arguments presented in &sect;2.6 of [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)]

[http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)] discuss linearization of the principal governing equations and stellar pulsation primarily from an Eulerian perspective.  Focusing on &sect;57 (pp. 455 - 458) of their ''Handbuch der Physik'' article &#8212; which falls under the major heading, "Radial oscillations of a gaseous sphere under its own gravitation" &#8212; we note, first that they use <math>~\delta r</math> to denote the radial displacement and use primes to identify all ''Eulerian'' perturbations.  Then, in separating out the spatial and time dependences, they use the notation (see their equation 57.14),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta r</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\delta r e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~i \sigma \delta r e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\rho^'(r) e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~p^'(r) e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Phi^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\Phi^'(r) e^{i\sigma t} \, .</math>
  </td>
</tr>
</table>
</div>
 
According to their derivations, the linearized equation of continuity (EOC), equation of motion (EOM), and Poisson equation &#8212; see, repectively, their equations 57.15, 57.16 (with nonadiabatic effects set to zero), and 57.21 &#8212; are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho^' + \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~- \sigma^2 \delta r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\partial \Phi^'}{\partial r} + \frac{\rho^'}{\rho^2} \frac{\partial p}{\partial r} - \frac{1}{\rho}\frac{\partial p^'}{\partial r} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2 \Phi^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho^' \, .</math>
  </td>
</tr>
</table>
</div>

Next, with the aim of writing each of the three terms on the right-hand-side of the EOM in terms of the displacement, <math>~\delta r</math>, Ledoux and Walraven do the following: (1) By combining the EOC with the Poisson equation they obtain the perturbation in the gravitational acceleration, namely (see their equation 57.22),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{r^2} \frac{\partial}{\partial r} \biggl( r^2 \frac{\partial\Phi^'}{\partial r} \biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( 4\pi G r^2 \rho \delta r \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ \frac{\partial\Phi^'}{\partial r}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4\pi G \rho \delta r \, .</math>
  </td>
</tr>
</table>
</div>

(2) The pressure fluctuation is obtained by using the adiabatic form of the 1<sup>st</sup> law of thermodynamics to relate <math>~P</math> to <math>~\rho</math>, replacing the ''Lagrangian'' time derivatives with their ''Eulerian'' counterparts, then using the linearized EOC to provide an expression for the density fluctuation in terms of the radial displacement.  Specifically (see their equations 56.15 and 57.19 with nonadiabatic terms set to zero),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \frac{d\rho}{dt}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ \frac{\partial p^'}{\partial t} + v^' \frac{\partial p}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \biggl[ \frac{\partial \rho^'}{\partial t} + v^' \frac{\partial \rho}{\partial r} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ p^'+ \delta r \frac{\partial p}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \biggl[ \rho^' + \delta r \frac{\partial \rho}{\partial r} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \biggl[ - \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr) + \delta r \frac{\partial \rho}{\partial r} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr] </math>
  </td>
</tr>
</table>
</div>
With these substitutions, the linearized EOM (tentatively) takes the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma^2 \rho\delta r - \rho \frac{\partial \Phi^'}{\partial r} - \frac{\partial p^'}{\partial r} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{\rho^'}{\rho} \frac{\partial p}{\partial r}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~~~ \rho \delta r \biggl[ \sigma^2  + 4\pi G \rho \biggr] 
+ \frac{\partial }{\partial r}\biggl\{ \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr]  
+\delta r \frac{\partial p}{\partial r} \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~~~ \rho \delta r \biggl[ \sigma^2  + 4\pi G \rho \biggr] 
+ \frac{\partial }{\partial r}\biggl\{ \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr]  \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr)\biggr]
- \frac{\partial}{\partial r} \biggl[ \delta r \frac{\partial p}{\partial r}  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{2\rho \delta r}{r} + \frac{\partial}{\partial r}(\rho \delta r)\biggr]
+ \biggl\{  \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl(\frac{2\rho\delta r}{r} \biggr)
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 \delta r \frac{\partial p}{\partial r} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl(\frac{4\rho\delta r}{r} \biggr) 
+\frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 \delta r \frac{\partial p}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~~~ \rho \delta r \biggl[ \sigma^2  + \frac{4Gm(r)}{r^3}\biggr] 
+ \frac{\partial }{\partial r}\biggl\{ \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr]  \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\rho\delta r(4\pi G \rho) 
+\frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 \delta r \frac{\partial p}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\rho\delta r \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl( r^2 \frac{\partial\Phi}{\partial r}\biggr) \biggr] 
+\frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 (\rho \delta r )\frac{1}{\rho}\frac{\partial p}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\rho\delta r \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl( r^2 \frac{\partial\Phi}{\partial r}\biggr) \biggr] 
- \frac{\partial \Phi}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
+ \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 (\rho \delta r )\frac{\partial \Phi}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
0 \, .
</math>
  </td>
</tr>
</table>
</div>
This matches equation (57.23) of Ledoux &amp; Walraven, if all nonadiabatic terms are set to zero.

==Separate Analyses of Homogeneous Sphere==
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 22 June 2015:  Last night I realized that a key to understanding how to reconcile the Eulerian and Lagrangian perspectives was an analysis of the eigenvalue problem for the homogeneous sphere.]]In an [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|accompanying discussion]], we have reviewed [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S T. E. Sterne's (1937, MNRAS, 97, 582)] study of radial pulsation modes in the homogeneous sphere.  He solved the eigenvalue problem as defined by the

<div align="center" id="LagragianApproach">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

and, hence, as established via a [[SSC/Perturbations#Consistent_Lagrangian_Formulation|Lagrangian formulation of the problem]].  The eigenvectors and eigenvalues that Sterne derived for the first two or three radial modes have also appeared &#8212; usually in the context of separate, re-derivations &#8212; in other publications:  See, for example, &sect;38.2 (pp. 402-403) of [[Appendix/References#KW94|[<font color="red">KW94</font>]]].

After finishing that review, we became aware that a separate study of radial pulsation modes in the homogeneous sphere has been published by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
In his book titled, ''The Pulsation Theory of Variable Stars'' (see, specifically his &sect; 3.2, beginning on p. 27).  Rosseland solved an eigenvalue problem as defined by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial r} \biggl( \gamma P_0 \nabla\cdot \vec{\xi}\biggr) + \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \rho_0 \xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
(see his equation 2.23 on p. 20, with the adiabatic condition being enforced by setting the right-hand-side equal to zero), where,
<div align="center">
<math>~\vec\xi = \mathbf{\hat{e}}_r \xi(r) \, .</math>
</div>

Rosseland derived this expression in an earlier section of his book via an Eulerian formulation of the problem.  
Realizing that, for a spherically symmetric system,
<div align="center">
<math>\nabla\cdot \vec\xi = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2 \xi\biggr) = \frac{\partial \xi}{\partial r} + \frac{2\xi}{r} \, ,</math>
</div>
as is demonstrated in [[SSC/Structure/OtherAnalyticModels#Eulerian_Approach|and accompanying discussion]], this relation can be rewritten in the more familiar form of a 2<sup>nd</sup>-order ODE, namely,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="left">
<math>~P_0 \frac{\partial^2 \xi}{\partial r^2}
+ \biggl[ \frac{2P_0}{r}- \rho_0 g_0  \biggr] \frac{\partial \xi}{\partial r}
+ \biggl[ \biggl( \frac{\omega^2\rho_c}{\gamma}  + \frac{4\rho_c g_0}{\gamma r}\biggr) \biggl(\frac{\rho_0}{\rho_c}\biggr)
- \biggl(\frac{2\rho_c g_0 }{r}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2P_0}{r^2} \biggr] \xi 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="right">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

Here we will repeat the setup and solution of both eigenvalue problems in an effort to reconcile differences.  As we have [[SSC/Stability/UniformDensity#Properties_of_the_Equilibrium_Configuration|explained elsewhere]], an equilibrium, homogeneous, self-gravitating sphere has the following structural properties:
<div align="center">
<math>\frac{\rho_0}{\rho_c} = 1 \, ,</math>
</div>
<div align="center">
<math>\frac{P_0}{P_c} = 1 - \chi_0^2 \, ,</math>
</div>
<div align="center">
<math> 
\frac{g_0}{g_\mathrm{SSC}} = 2\chi_0  \, ,
</math><br />
</div>
where,
<div align="center">
<math>P_c = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) </math>
&nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  and &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\rho_c = \frac{3M}{4\pi R^3} \, ,</math>
</div>
and the characteristic gravitational acceleration is defined as,
<div align="center">
<math> 
g_\mathrm{SSC} \equiv \frac{P_c}{R \rho_c} \, .
</math>
</div>
In addition, it will be useful to recognize that the square of the characteristic time for dynamical oscillations in spherically symmetric configurations (SSC) is,
<div align="center">
<math> 
\tau_\mathrm{SSC}^2 \equiv \frac{R^2 \rho_c}{P_c}  = \frac{2R^3}{G M} \, .
</math><br />
</div>


==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>

==Eulerian Reformulation==

Using the same characteristic time scale and gravitational acceleration, we can similarly rewrite the ''Eulerian-formulated'' expression as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="left">
<math>~
\biggl(\frac{P_0}{P_c}\biggr) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \biggl[ \frac{2}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr)  - \frac{g_0 }{g_\mathrm{SSC}}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \biggl[\frac{\omega^2\tau_\mathrm{SSC}^2}{\gamma}  + \frac{2}{\chi_0 } \biggl(\frac{2}{\gamma } - 1\biggr)\frac{g_0}{g_\mathrm{SSC}}\biggr] 
\biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2}{\chi_0^2} \biggl(\frac{P_0}{P_c}\biggr)  \biggr\} \xi
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="right">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, [[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|"linear stellar model"]] are plugged into it, we obtain exactly the same 2<sup>nd</sup>-order ODE as published by [http://adsabs.harvard.edu/abs/1967ApJ...148..305S R. Stothers &amp; J. A. Frogel (1967, ApJ, 148, 305)] &#8212; see their equation (2).  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{\partial^2 \xi}{\partial \chi_0^2}
+ \biggl[ \frac{2( 1-\chi_0^2 )}{\chi_0}  - 2\chi_0\biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \frac{1}{\gamma}\biggl[2\biggl( \frac{\omega^2 R^3}{GM} \biggr) + 4\biggl(2 - \gamma\biggr) \biggr] 
- \frac{2}{\chi_0^2} ( 1-\chi_0^2 )  \biggr\} \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{1}{\chi_0}\biggl[ 2 - 4\chi_0^2\biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \frac{2}{\gamma}\biggl[\biggl( \frac{\omega^2 R^3}{GM} \biggr) + \biggl(4 - 3\gamma\biggr) +\gamma\biggr] 
- \frac{2}{\chi_0^2} ( 1-\chi_0^2 )  \biggr\} \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{1}{\chi_0}\biggl[ 4 - 6\chi_0^2 - 2(1-\chi_0^2) \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \mathfrak{F} + \biggl(4 - \frac{2}{\chi_0^2} \biggr) \biggr\} \xi \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{1}{\chi_0}\biggl[ 4 - 6\chi_0^2 \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \mathfrak{F} \xi 
- \frac{1}{\chi_0}\biggl[ 2(1-\chi_0^2) \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl(4 - \frac{2}{\chi_0^2} \biggr) \xi 
</math>
  </td>
</tr>
</table>
</div>


Drawing from Rosseland's presentation (see his p. 29), the following table details the eigenfunctions for the three lowest radial modes that satisfy this wave equation; a virtually identical summary can be found in &sect;6.5 (see especially Figure 6.1  on p. 140) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>].

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="5">
Rosseland's (1964) Eigenfunctions for Homogeneous Sphere<p></p>
Figure in the right-most column extracted from p. 29 of [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)]<p></p>
"''The Pulsation Theory of Variable Stars''" (New York:  Dover Publications, Inc.)
  </th>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center" colspan="2">Eigenvector</td>
  <td align="center">Square of Eigenfrequency:<p></p><math>~3\sigma^2/(4\pi G\rho)</math></td>
  <td align="center" rowspan="5">[[File:RosselandModesFigure3.png|350px|center|Rosseland (1937)]]</td>
</tr>
<tr>
  <td align="center"><math>~m</math></td>
  <td align="center">As Published</td><td align="center">Rewritten</td>
  <td align="center"><math>~\frac{m}{2}(m+1)\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="left"><math>~\xi = -2\chi_0</math></td>
  <td align="left"><math>~-\frac{1}{2}\biggl( \frac{\xi}{\chi_0}\biggr) = 1</math></td>
  <td align="center"><math>~3\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~4</math></td>
  <td align="left"><math>~\xi = -\frac{20}{3}\chi_0 + \frac{28}{3}\chi_0^3</math></td>
  <td align="left"><math>~-\frac{3}{20}\biggl( \frac{\xi}{\chi_0}\biggr) = 1- \frac{7}{5}\chi_0^2</math></td>
  <td align="center"><math>~10\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~6</math></td>
  <td align="left"><math>~\xi = -14\chi_0 + \frac{252}{5} \chi_0^3 - \frac{198}{5} \chi_0^5</math></td>
  <td align="left"><math>~-\frac{1}{14}\biggl( \frac{\xi}{\chi_0}\biggr) = 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>
</table>
</div>

==Reconciliation by Rosseland==
[https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1969)] points out that these two different formulations can be reconciled by adopting the following dependent coordinate substitution (see his equation 3.11):
<div align="center">
<math>~x \rightarrow \frac{\xi}{\chi_0} \, .</math>
</div>
This means that the derivatives that appear in the above ''Lagrangian formulation'' should be replaced by the expressions,
<div align="center">
<math>~\frac{dx}{d\chi_0} \rightarrow \biggl(\frac{1}{\chi_0}\biggr)\frac{d\xi}{d\chi_0} - \frac{\xi}{\chi_0^2} \, ,</math>
</div>
and,
<div align="center">
<math>~\frac{d^2x}{d\chi_0^2} \rightarrow \biggl(\frac{1}{\chi_0}\biggr)\frac{d^2\xi}{d\chi_0^2} - \biggl( \frac{2}{\chi_0^2} \biggr) \frac{d\xi}{d\chi_0} 
+ \frac{2\xi}{\chi_0^3} \, .</math>
</div>
Doing this, we have,

<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 )\biggl[ \biggl(\frac{1}{\chi_0}\biggr)\frac{d^2\xi}{d\chi_0^2} - \biggl( \frac{2}{\chi_0^2} \biggr) \frac{d\xi}{d\chi_0} 
+ \frac{2\xi}{\chi_0^3}  \biggr] + \frac{1}{\chi_0}\biggl[4 -  6\chi_0^2 \biggr] \biggl[ \biggl(\frac{1}{\chi_0}\biggr)\frac{d\xi}{d\chi_0} 
- \frac{\xi}{\chi_0^2} \biggr] + \mathfrak{F} \biggl( \frac{\xi}{\chi_0} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{1}{\chi_0}\biggr) \biggl\{
( 1-\chi_0^2 )\biggl[ \frac{d^2\xi}{d\chi_0^2} - \biggl( \frac{2}{\chi_0} \biggr) \frac{d\xi}{d\chi_0} 
+ \frac{2\xi}{\chi_0^2}  \biggr] + \biggl[4 -  6\chi_0^2 \biggr] \biggl[ \biggl(\frac{1}{\chi_0}\biggr)\frac{d\xi}{d\chi_0} 
- \frac{\xi}{\chi_0^2} \biggr] + \mathfrak{F} \xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{1}{\chi_0}\biggr) \biggl\{
( 1-\chi_0^2 )\frac{d^2\xi}{d\chi_0^2} + \frac{1}{\chi_0} \biggl[4 -  6\chi_0^2 \biggr] \frac{d\xi}{d\chi_0} + \mathfrak{F} \xi 
-\biggl( \frac{2}{\chi_0} \biggr)  ( 1-\chi_0^2 )\frac{d\xi}{d\chi_0}  + \biggl[4\chi_0^2 -2 \biggr] \frac{\xi}{\chi_0^2}  \biggr\}  \, ,
</math>
  </td>
</tr>
</table>
</div>
which, indeed, matches the expression derived via Rosseland's ''Eulerian formulation''.

=Rosseland's Derivation of the Wave Equation=

Here we step through a derivation of the wave equation as has been presented by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
in chapter 2 of his book titled, ''The Pulsation Theory of Variable Stars'' (see, especially his &sect;&sect; 2.1 &amp; 2.2, pp. 15-20), dropping all terms that account for nonadiabatic effects.

We begin with the set of [[PGE#Principal_Governing_Equations|principal governing equations]] that provides the foundation for practically all of our discussions in this H_Book, namely, the

<div align="center">
<span id="PGE:Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

{{Math/EQ_Continuity01}}



<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />

{{Math/EQ_Euler01}}


<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{Math/EQ_FirstLaw02}} .


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{Math/EQ_Poisson01}}
</div>


Because we are considering only spherically symmetric configurations, we will implement operators that reflect structural variations only in the radial coordinate.  Specifically, following a [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|parallel discussion of the principal governing equations that are appropriate for spherically symmetric configurations]], we have the,

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{\partial (r^2 v_r)}{\partial r}  \biggr] = 
\frac{d\rho}{dt} + \rho \biggl[ \frac{\partial v_r}{\partial r} + \frac{2v_r}{r} \biggr] = 
0 </math>



<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </math><br />



<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{Math/EQ_FirstLaw02}}


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math><br />
</div>

==Step 1:==
Following Rosseland, we adopt as a supplemental relation what we have [[SR/IdealGas#Consequential_Ideal_Gas_Relations|referred to elsewhere]] as,

<div align="center">
<span id="ConservingMass:Lagrangian"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation of State,

{{Math/EQ_EOSideal02}}
</div>
where, for our purposes, we assume that {{Math/MP_AdiabaticIndex}} is independent of space and time.  Plugging the function, <math>~\epsilon(P,\rho)</math>, that is defined by this equation of state into the adiabatic form of the 1<sup>st</sup> law of thermodynamics leads to the relations (see, respectively, Rosseland's equations 2.8 and 2.9),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_g P}{\rho} \frac{d\rho}{dt} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{P}{P_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_0}\biggr)^{\gamma_g} \, ,</math>
  </td>
</tr>
</table>
</div>

where <math>~P_0</math> and <math>~\rho_0</math> are independent of time.  We note for later use that, when the equation of continuity &#8212; as written in its original vector-operator form &#8212; is combined with this last differential form of the 1<sup>st</sup> law of thermodynamics, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma_g P \nabla\cdot \vec{v} \, .</math>
  </td>
</tr>
</table>
</div>


==Step 2:==
As we have shown in a 
[[SSCpt2/SolutionStrategies#Technique_1|separate discussion of how to construct equilibrium, spherically symmetric configurations]], the Poisson equation can be integrated once to give an expression for the gravitational acceleration (see, also, Rosseland's equation 2.16) of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g \equiv \frac{d\Phi}{dr}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_r}{r^2} \, .</math>
  </td>
</tr>
</table>
</div>

As is quite customary in this field of research, Rosseland substitutes <math>~g</math> for the radial gradient of the gravitational potential in the Euler equation &#8212; that is, he writes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dv_r}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho} \frac{\partial P}{\partial r} - g \, ,</math>
  </td>
</tr>
</table>
</div>

(see his equation 2.14 with nonadiabatic terms set to zero) &#8212; and, in so doing, considers that the Poisson equation has been solved.  This reduces the number of governing equations (as well as the number of unknown structural variables) to three.  Then Rosseland proceeds to point out that, when the radial pulsation of a spherically symmetric configuration is viewed from a Lagrangian perspective &#8212; that is, while riding along with a fluid element &#8212; the mass enclosed inside the radial position of each fluid element, <math>~M_r</math>, does not change with time.  Hence, the ''Lagrangian'' time-derivative of the gravitational acceleration is (see Rosseland's equation 2.17),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dg}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~GM_r \frac{d}{dt}\biggl(\frac{1}{r^2}\biggr) = - \frac{2GM_r}{r^3} \frac{dr}{dt} = - \frac{2gv_r}{r} \, .</math>
  </td>
</tr>
</table>
</div>

==Step 3:==
Next, Rosseland takes the ''Lagrangian'' time-derivative of the Euler equation, which gives (see Rosseland's equation 2.15 with nonadiabatic terms set to zero),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{d}{dt}\biggl[\frac{1}{\rho} \frac{\partial P}{\partial r}\biggr] - \frac{dg}{dt} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\rho^2}\frac{d\rho}{dt} \biggl(\frac{\partial P}{\partial r}\biggr)
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] - \frac{dg}{dt} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho}\biggl(\frac{\partial P}{\partial r}\biggr)\biggl[ \frac{\partial v_r}{\partial r} + \frac{2v_r}{r}  \biggr]
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] + \frac{2gv_r}{r} \, .</math>
  </td>
</tr>
</table>
</div>
But, from the Euler equation, itself, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{1}{\rho} \biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{dv_r}{dt} + g \, .</math>
  </td>
</tr>
</table>
</div>
Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{dv_r}{dt} + g \biggr] \biggl[ \frac{\partial v_r}{\partial r} + \frac{2v_r}{r}  \biggr]
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] + \frac{2gv_r}{r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{dv_r}{dt} +g\biggr]\frac{\partial v_r}{\partial r}  
- \frac{1}{\rho} \frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

==Step 4:==
Finally, Rosseland sets out to swap the order of the time- and space-derivatives in the term,
<div align="center">
<math>~\frac{d}{dt}\biggl[\frac{\partial P}{\partial r}\biggr] \, .</math>
</div>
But, whereas the ''partial'' time derivative commutes with the ''partial'' radial derivative, the total time derivative does not commute with the spatial derivative.  Generically, however, for any scalar variable, <math>~q</math>, we can swap between total and partial time derivatives via the operator relation,
<div align="center">
<math>~\frac{dq}{dt} = \frac{\partial q}{\partial t} + v_r \frac{\partial q}{\partial r} \, .</math>
</div>
Hence, if <math>~q</math> is replaced by <math>~P</math>, we can write,
<div align="center">
<math>~ v_r \biggl(\frac{\partial P}{\partial r}\biggr)  = \frac{dP}{dt}  - \frac{\partial P}{\partial t}\, ;</math>
</div>
while, if <math>~q</math> is replaced by <math>~\partial P/\partial r</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial t}\biggl( \frac{\partial P}{\partial r} \biggr) + v_r \frac{\partial }{\partial r} \biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ v_r \biggl( \frac{\partial P}{\partial r} \biggr)\biggr] 
- \biggl( \frac{\partial P}{\partial r} \biggr)\frac{\partial v_r}{\partial r} \, ,</math>
  </td>
</tr>
</table>
</div>
where, in making this last step, we have swapped the order of the ''partial'' derivatives in the first term on the right-hand side.  Combining these two relations and incorporating the form of Euler's equation that was highlighted in step #3 gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl( \frac{\partial P}{\partial r} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  - \frac{\partial P}{\partial t} \biggr] 
+ \rho \biggl[  \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] 
+ \rho \biggl[  \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r} \, .</math>
  </td>
</tr>
</table>
</div>
Inserting this into the equation found at the end of step #3, then inserting the expression for <math>~dP/dt</math> derived in step #1 gives, 

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \gamma_g P \nabla\cdot \vec{v}\biggr] + \frac{4gv_r}{r}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
This matches Rosseland's equation (2.18) with the nonadiabatic terms set to zero.




{{ SGFworkInProgress }}

=Linearizing the Key Relations=

<table border="3" align="center" cellpadding="10">
<tr>
  <th align="center" colspan="2"><font color="darkgreen" size="+1">Continuity Equation</font></th>
</tr>
<tr>
  <th align="center" width="50%">Lagrangian Perspective</th>
  <th align="center" width="50%">Eulerian Perspective</th>
</tr>
<tr>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\rho}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \rho \nabla\cdot\vec{v}</math>
  </td>
</tr>
</table>
</div>

  </td>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial\rho}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \rho \nabla\cdot\vec{v} - \vec{v}\cdot \nabla\rho</math>
  </td>
</tr>
</table>
</div>

  </td>
</tr>
<tr>
  <th align="center" colspan="2">Spherically Symmetric Initial Configurations &amp; Purely Radial Perturbations</th>
</tr>
<tr>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\rho}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr)</math>
  </td>
</tr>
</table>
</div>

  </td>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial\rho}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr) - v_r \frac{\partial \rho}{\partial r}</math>
  </td>
</tr>
</table>
</div>

  </td>
</tr>
<!-- EXPLAIN PERTURBATION VARIABLE CHOICES -->
<tr>
  <td align="left" colspan="2">
In an interval of time, <math>~dt = \partial t</math>, a fluid element initially at position <math>~r_0</math> moves to position, <math>~r = r_0 + r_1 = r_0(1 + \xi)</math>.  [For later reference, note that <math>~\xi</math> can be a function of <math>~r_0</math> as well as of <math>~t</math>.]  On the righthand side of the expression, the radial coordinate will be handled as follows:  From the Lagrangian perspective, <math>~r \rightarrow r_0 (1+ \xi)</math>, while from the Eulerian perspective, we want to stay at the original coordinate location, so <math>~r \rightarrow r_0</math>.  From both perspectives, 
<div align="center"><math>~v_r = \frac{\partial ( r_0 \xi )}{\partial t} = r_0 \frac{\partial \xi}{\partial t}  \, .</math></div>
Riding with the fluid element (Lagrangian perspective), <math>~\rho \rightarrow (\rho_0 + \rho_L) = \rho_0(1+s_L)</math>, while at a fixed coordinate location (Eulerian perspective), <math>~\rho \rightarrow (\rho_0 + \rho_E) = \rho_0(1 + s_E)</math>.  Finally, in maintaining a ''Lagrangian'' perspective, we will need to ensure that the same element of mass is being tracked as we "ride along" with the fluid element to its new position.  For radial perturbations associated with a spherically symmetric configuration, this means that the differential mass in each spherical shell, <math>~dm = 4\pi r^2 \rho dr</math>, must remain constant; that is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~4\pi r_0^2 \rho_0 dr_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r^2 \rho dr</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi [r_0(1+\xi)]^2 \rho_0(1+s_L) dr</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r_0^2 \rho_0 \biggl(1+2\xi + \cancelto{\scriptstyle\text{small}}{\xi^2} + \cdots \biggr) (1+s_L) dr</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~4\pi r_0^2 \rho_0 \biggl(1+2\xi + s_L + 2\cancelto{\scriptstyle\mathrm{small}}{\xi s_L} \biggr) dr</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{d}{dr}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~(1+2\xi + s_L ) \frac{d}{dr_0} \, .</math>
  </td>
</tr>
</table>
</div>
  </td>
</tr>

<tr>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl[\rho_0(1+s_L)\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl\{ \frac{\rho_0(1+\cancelto{}{s_L})}{[r_0(1+\cancelto{}{\xi})]^2} \biggr\}(1+2\cancelto{}{\xi s_L}) \frac{\partial}{\partial r_0} 
\biggl\{ [r_0(1+\cancelto{}{\xi})]^2 v_r \biggr\}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{d s_L}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\frac{2v_r}{r_0} + \frac{\partial v_r}{\partial r_0} \biggr] - \frac{1}{\rho_0} \frac{d\rho_0}{dt}</math>
  </td>
</tr>
</table>
</div>

  </td>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial}{\partial t}\biggl[\rho_0(1+s_E)\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho_0(1+\cancelto{\mathrm{small}}{s_E})}{r_0^2} \frac{\partial}{\partial r_0} \biggl( r_0^2 v_r \biggr) 
- v_r \frac{\partial [\rho_0(1+\cancelto{\mathrm{small}}{s_E})] }{\partial r_0}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{\partial s_E}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\frac{2v_r}{r_0} + \frac{\partial v_r}{\partial r_0} \biggr] 
- \frac{v_r}{\rho_0} \frac{\partial \rho_0 }{\partial r_0}</math>
  </td>
</tr>
</table>
</div>

  </td>
</tr>
<tr>
  <td align="left" colspan="2">
Note:  The last term that appears on the righthand side of the two expressions appears to be different.  But if, as we are assuming here, <math>~\rho_0</math> has no explicit time dependence but ''may'' be considered to be a function of the radial coordinate, <math>~r_0</math>, then the two terms are the same.  This is because, quite generically for any scalar function <math>~q</math>, the ''total'' time-derivative (Lagrangian perspective) differs from the ''partial'' time-derivative (Eulerian perspective) via the expression, <math>dq/dt - \partial q /\partial t = \vec{v}\cdot \nabla q</math>.  In our case, <math>~\partial \ln \rho_0/\partial t = 0</math>, so <math>~d\ln\rho_0/dt = \vec{v}\cdot \nabla \ln \rho_0</math>.
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
<math>~s_L ~~\rightarrow~~ \Delta_L(r_0) e^{i\omega t}</math>
&nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  &hellip; and &hellip; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  
<math>~s_E ~~\rightarrow~~ \Delta_E(r_0) e^{i\omega t}</math>

<math>~\xi ~~\rightarrow~~ x(r_0) e^{i\omega t}</math>
&nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  
<math>~v_r ~~\rightarrow~~ (i\omega)r_0 x(r_0) e^{i\omega t}</math>
  </td>
</tr>

<tr>
  <td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{i\omega t} \biggl[ (i\omega)\Delta_L + \frac{d\Delta_L}{dt} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- e^{i\omega t} \biggl[ 2(i\omega)x + (i\omega)x +  (i\omega)r_0 \frac{\partial x}{\partial r_0} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~r_0 \frac{\partial x}{\partial r_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_L -3 x - \frac{1}{(i\omega)} \biggl[ v_r \frac{\partial\Delta_L}{\partial r_0} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_L - x \biggl[3 + \cancelto{\scriptstyle\mathrm{small}}{\frac{\partial\Delta_L}{\partial \ln r_0} }\biggr]</math>
  </td>
</tr>
</table>
</div>
  </td>
  <td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{i\omega t} (i\omega)\Delta_E </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- e^{i\omega t} \biggl[ 2(i\omega)x + (i\omega)x +  (i\omega)r_0 \frac{\partial x}{\partial r_0} \biggr]
- \biggl[\frac{1}{\rho_0} \frac{\partial \rho_0}{\partial r_0} \biggr](i\omega)r_0 x e^{i\omega t}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~r_0 \frac{\partial x}{\partial r_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_E - x \biggl[3 + \frac{\partial \ln\rho_0}{\partial \ln r_0} \biggr] </math>
  </td>
</tr>
</table>
</div>
  </td>
</tr>
</table>

=See Also=

* [[SSC/StabilityEulerianPerspective|Eulerian Perspective]]
* [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Lagrangian Perspective]]

* Especially in the context of our [[#Approach_by_Ledoux_and_Walraven|above review]] of the approach presented by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)], it also might prove valuable to look at the arguments presented in &sect;2.6 of [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)]

* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell &amp; J. P. Ostriker (1967)], MNRAS, 136, 293:  ''On the stability of differentially rotating bodies'' &#8212; this article not only develops a variational principle from both an Eulerian and Lagrangian perspective, it also develops an ''anti-spiral theorem'' that might be relevant in the context of Cases' compressible analog of Riemann ellipsoids.

{{ SGFfooter }}