User:Tohline/SSC/Perspective Reconciliation - Revision history

Tohline: /* See Also */

2019-06-28T23:27:40Z

Tohline: /* See Also */

2019-06-28T23:14:52Z

Tohline: /* See Also */

2019-06-28T23:06:25Z

Tohline at 20:13, 26 June 2019

2019-06-26T20:13:19Z

← Older revision		Revision as of 20:13, 26 June 2019
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	=Reconciling Eulerian versus Lagrangian Perspectives=		=Reconciling Eulerian versus Lagrangian Perspectives=
	Note: It also might prove valuable to look at the arguments presented in §2.6 of [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)]


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	===Approach by Ledoux and Walraven===		===Approach by Ledoux and Walraven===
			Note: It also might prove valuable to look at the arguments presented in §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 & Th. Walraven (1958)] discuss linearization of the principal governing equations and stellar pulsation primarily from an Eulerian perspective. Focusing on §57 (pp. 455 - 458) of their ''Handbuch der Physik'' article — which falls under the major heading, "Radial oscillations of a gaseous sphere under its own gravitation" — 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),		[http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] discuss linearization of the principal governing equations and stellar pulsation primarily from an Eulerian perspective. Focusing on §57 (pp. 455 - 458) of their ''Handbuch der Physik'' article — which falls under the major heading, "Radial oscillations of a gaseous sphere under its own gravitation" — 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">		<div align="center">

Tohline: /* Reconciling Eulerian versus Lagrangian Perspectives */

2019-06-26T20:11:10Z

Reconciling Eulerian versus Lagrangian Perspectives

← Older revision		Revision as of 20:11, 26 June 2019
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	=Reconciling Eulerian versus Lagrangian Perspectives=		=Reconciling Eulerian versus Lagrangian Perspectives=
			Note: It also might prove valuable to look at the arguments presented in §2.6 of [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)]


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Tohline: /* Eulerian Reformulation */ Insert reference to parallel discussion in ST83

2016-09-30T14:46:32Z

Eulerian Reformulation: Insert reference to parallel discussion in ST83

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	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.		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 §6.5 (see especially Figure 6.1 on p. 140) of [<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>].

	<div align="center">		<div align="center">

Tohline: /* Reconciling Eulerian versus Lagrangian Perspectives */

2015-06-29T17:04:55Z

Reconciling Eulerian versus Lagrangian Perspectives

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	</tr>		</tr>
	<tr>		<tr>
	<td align="left" colspan="2"><sup>&dagger;</sup>See especially §43 (p. 432) and §55 (p. 452) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)]~~. Particularly clarifying comments from this reference are the following~~:		<td align="left" colspan="2"><sup>&dagger;</sup>See especially the following clarifying comments from, respectively, §43 (p. 432) and §55 (p. 452) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & 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,		* 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,

Tohline: /* Reconciling Eulerian versus Lagrangian Perspectives */

2015-06-28T17:14:03Z

Reconciling Eulerian versus Lagrangian Perspectives

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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\hat{e}_r [0 + v_r^'(r) e^{i\sigma t}] = i\sigma \delta r(r) e^{i\sigma t} </math>		<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>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\hat{e}_r [0 + \delta v_r(m) e^{i\sigma t}] = i\sigma \delta r(m) e^{i\sigma t}</math>		<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>		</td>
	</tr>		</tr>

Tohline: /* Reconciling Eulerian versus Lagrangian Perspectives */ Add e^(i\sigma t) to all perturbed quantities in table

2015-06-28T17:11:32Z

Reconciling Eulerian versus Lagrangian Perspectives: Add e^(i\sigma t) to all perturbed quantities in table

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	<table border="1" cellpadding="8" align="center" width="90%">		<table border="1" cellpadding="8" align="center" width="90%">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\hat{e}_r (r_0 + \delta r) </math>		<math>~\hat{e}_r [r_0 + \delta r(r) e^{i\sigma t}] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\hat{e}_r (0 + v_r^') </math>		<math>~\hat{e}_r [0 + v_r^'(r) e^{i\sigma t}] = i\sigma \delta r(r) e^{i\sigma t} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\rho_0 + \rho^' </math>		<math>~\rho_0 + \rho^'(r) e^{i\sigma t} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~P_0 + P^' </math>		<math>~P_0 + P^'(r) e^{i\sigma t} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\Phi_0 + \Phi^' </math>		<math>~\Phi_0 + \Phi^'(r) e^{i\sigma t} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\hat{e}_r (r_0 + \delta r) </math>		<math>~\hat{e}_r [r_0 + \delta r(m) e^{i\sigma t}] </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\hat{e}_r (0 + \delta v_r) </math>		<math>~\hat{e}_r [0 + \delta v_r(m) e^{i\sigma t}] = i\sigma \delta r(m) e^{i\sigma t}</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\rho_0 + \delta \rho </math>		<math>~\rho_0 + \delta \rho(m) e^{i\sigma t} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~P_0 + \delta P </math>		<math>~P_0 + \delta P(m) e^{i\sigma t} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\Phi_0 + \delta \Phi </math>		<math>~\Phi_0 + \delta \Phi(m) e^{i\sigma t} </math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Reconciling Eulerian versus Lagrangian Perspectives */ Introduce table that clarifies the notation adopted from LW58

2015-06-28T16:49:21Z

Reconciling Eulerian versus Lagrangian Perspectives: Introduce table that clarifies the notation adopted from LW58

← Older revision		Revision as of 16:49, 28 June 2015
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	=Reconciling Eulerian versus Lagrangian Perspectives=		=Reconciling Eulerian versus Lagrangian Perspectives=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

			<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 & 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">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\hat{e}_r (r_0 + \delta r) </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\hat{e}_r v</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\hat{e}_r (0 + v_r^') </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\rho</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\rho_0 + \rho^' </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~P</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~P_0 + P^' </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\Phi</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\Phi_0 + \Phi^' </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">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\hat{e}_r (r_0 + \delta r) </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\hat{e}_r v</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\hat{e}_r (0 + \delta v_r) </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\rho</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\rho_0 + \delta \rho </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~P</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~P_0 + \delta P </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\Phi</math>
			</td>
			<td align="center">
			<math>~\rightarrow</math>
			</td>
			<td align="left">
			<math>~\Phi_0 + \delta \Phi </math>
			</td>
			</tr>
			</table>

			</td>
			</tr>
			<tr>
			<td align="left" colspan="2"><sup>&dagger;</sup>See especially §43 (p. 432) and §55 (p. 452) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux & Walraven (1958)]. Particularly clarifying comments from this reference are the following:
			* 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==		==Introduction==

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	* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies'' — 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.		* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies'' — 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.


			{{LSU_HBook_footer}}

← Older revision		Revision as of 23:14, 28 June 2019
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	* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Lagrangian Perspective]]		* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Lagrangian Perspective]]

	~~Note~~: It also might prove valuable to look at the arguments presented in §2.6 of [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)]		* 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 & Th. Walraven (1958)], it also might prove valuable to look at the arguments presented in §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~~ & Th. ~~Walraven~~ (~~1958~~)] ~~discuss linearization of the principal governing equations and stellar pulsation primarily from an Eulerian perspective.~~ ~~Focusing on §57 (pp. 455 - 458) of their~~ ''~~Handbuch der Physik~~'' ~~article~~ — ~~which falls under the major heading, "Radial oscillations of~~ a ~~gaseous sphere under its own gravitation" — 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),~~		* [https://ui.adsabs.harvard.edu/abs/1967MNRAS.136..293L/abstract D. Lynden-Bell & J. P. Ostriker (1967)], MNRAS, 136, 293: ''On the stability of differentially rotating bodies'' — 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.

← Older revision		Revision as of 23:06, 28 June 2019
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	* [[User:Tohline/SSC/Stability_Eulerian_Perspective\|Eulerian Perspective]]		* [[User:Tohline/SSC/Stability_Eulerian_Perspective\|Eulerian Perspective]]
	* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Lagrangian Perspective]]		* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Lagrangian Perspective]]

			Note: It also might prove valuable to look at the arguments presented in §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 & Th. Walraven (1958)] discuss linearization of the principal governing equations and stellar pulsation primarily from an Eulerian perspective. Focusing on §57 (pp. 455 - 458) of their ''Handbuch der Physik'' article — which falls under the major heading, "Radial oscillations of a gaseous sphere under its own gravitation" — 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),