AxisymmetricConfigurations/SolvingPE - Revision history

Jet53man: /* Related Discussions */

2021-09-13T22:37:40Z

Related Discussions

← Older revision		Revision as of 18:37, 13 September 2021
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	<li>		<li>
	(Apollonian Circles) [[2DStructure/ToroidalCoordinateIntegrationLimits\|Toroidal-Coordinates Integration Limits]]		(Apollonian Circles) [[2DStructure/ToroidalCoordinateIntegrationLimits\|Toroidal-Coordinates Integration Limits]]
			</li>
			<li>
			[[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Configurations_and_Related_Coordinate_Systems\|Determining the Relationships between Toroidal Coordinates and other Coordinate Systems]]
	</li>		</li>
	</ul>		</ul>

Jet53man: /* Related Discussions */

2021-09-13T21:42:49Z

Related Discussions

← Older revision		Revision as of 17:42, 13 September 2021
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	=Related Discussions=		=Related Discussions=
			<ul>
			<li>Common Theme:
			<ul>
			<li>
			Menu Tile:  [[AxisymmetricConfigurations/SolvingPE\|Solving the (Axisymmetric) Poisson Equation]]
			<math>\Rightarrow</math>     Points principally to: [[AxisymmetricConfigurations/PoissonEq\|AxisymmetricConfigurations/PoissonEq]]
			</li>
			<li>
			Menu Tile:  [[2DStructure/UsingTC\|Using Toroidal Coordinates to Determine the Gravitational Potential]]
			<math>\Rightarrow</math>     Points principally to: [[2DStructure/ToroidalGreenFunction\|2DStructure/ToroidalGreenFunction]]
			</li>
			<li>
			Menu Tile:  [[2DStructure/TCsimplification\|Attempt at Simplification]]
			<math>\Rightarrow</math>     Points principally to: [[2DStructure/ToroidalCoordinates\|2DStructure/ToroidalCoordinates]]
			</li>
			<li>
			Menu Tile:  [[Apps/WongAP\|Wong's Analytic Potential (1973)]]
			<math>\Rightarrow</math>     Points principally to: [[Apps/Wong1973Potential\|Apps/Wong1973Potential]]
			</li>
			<li>
			Menu Tile:  [[Apps/DWT\|Dyson-Wong Tori]] (Thin Ring Approximation)
			<math>\Rightarrow</math>     Points principally to: [[Apps/DysonWongTori\|Apps/DysonWongTori]]
			</li>
			</ul>
			</li>
			<li>
			(Apollonian Circles) [[2DStructure/ToroidalCoordinateIntegrationLimits\|Toroidal-Coordinates Integration Limits]]
			</li>
			</ul>


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Jet53man at 17:21, 10 September 2021

2021-09-10T17:21:02Z

← Older revision		Revision as of 13:21, 10 September 2021
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	<td align="left" colspan="2">		<td align="left" colspan="2">
	{\| class="~~wikitable~~" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="Chap20A" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[~~User:Tohline/~~Apps/DysonWongTori#Thin_Ring_Approximation\|Dyson-Wong<br /> Tori]]<br />(Thin Ring Approximation)		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[Apps/DysonWongTori#Thin_Ring_Approximation\|Dyson-Wong<br /> Tori]]<br />(Thin Ring Approximation)
	\|}		\|}
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	<td align="left" colspan="2">		<td align="left" colspan="2">
	{\| class="~~wikitable~~" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="Chap20B" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:#D0FFFF;" \|[[~~User:Tohline/~~AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically\|Solving the<br /> Poisson Equation]]		! style="height: 150px; width: 150px; background-color:#D0FFFF;" \|[[AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically\|Solving the<br /> Poisson Equation]]
	\|}		\|}
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	<td align="left" colspan="2">		<td align="left" colspan="2">
	{\| class="~~wikitable~~" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="Chap20C" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Attempt at<br />Simplification]]		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Attempt at<br />Simplification]]
	\|}		\|}
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	{\| class="~~wikitable~~" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="Chap20D" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[~~User:Tohline/~~2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Using<br /> Toroidal Coordinates<br /> to Determine the<br /> Gravitational<br /> Potential]]		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Using<br /> Toroidal Coordinates<br /> to Determine the<br /> Gravitational<br /> Potential]]
	\|}		\|}
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	{\| class="~~wikitable~~" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="Chap20E" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[~~User:Tohline/~~Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|Wong's <br />(1973) <br >Analytic Potential]]		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|Wong's <br />(1973) <br >Analytic Potential]]
	\|}		\|}
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Jet53man at 17:11, 10 September 2021

2021-09-10T17:11:45Z

@@ Line 2: / Line 2: @@
 <!-- __NOTOC__ will force TOC off -->
 =Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions=
 You have arrived at this page from our [[H_BookTiledMenu#Axisymmetric_Equilibrium_Structures|''Tiled Menu'']] by clicking on the chapter title that is also identified, below, by the light-blue-colored icon.  You may proceed directly to that chapter by clicking (again) on that highlighted chapter icon.  However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected.  The common thread is the "Key Equation" titled, ''Gravitational Potential of an Axisymmetric Mass Distribution'', that appears at the top of the following set of chapter synopses.
 &nbsp;<br />
-=See Also=
+==Chapter Synopses==
-* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] &#8212; ''On the Dynamics of Revolving Fluids''
-* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, &amp; S. A. Teukolsky (1977, ApJ, 214, 584- 597)] &#8212; ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)
+===Part I===
-<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press &amp; Teukolsky make the following statement: "&hellip; we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincar&eacute;-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) &hellip;"</li></ol>
+{{Template:LSU_CT99CommonTheme1 }}
-* Referring to our [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|accompanying discussion of Type 1 Riemann ellipsoids]], it seems that uniform rotation on cylinders is not required.  What's going on?
 {{ SGFfooter }}

Jet53man at 17:07, 10 September 2021

2021-09-10T17:07:51Z

← Older revision		Revision as of 13:07, 10 September 2021
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	__FORCETOC__		__FORCETOC__
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=~~Solving~~ the ~~Poisson Equation~~=		=Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions=
	{\| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:white;" \|[[H_BookTiledMenu#Two-Dimensional_Configurations_.28Axisymmetric.29\|<b>Solving the<br />Poisson Equation</b>]]		! style="height: 150px; width: 150px; background-color:white;" \|[[H_BookTiledMenu#Two-Dimensional_Configurations_.28Axisymmetric.29\|<b>Solving the<br />Poisson Equation</b>]]
	\|}		\|}
	~~When constructing rotating equilibrium configurations that obey a~~ [[SR#~~Time-Independent_Problems~~\|~~barotropic equation of state~~]], ~~keep~~ in ~~mind that certain physical variable profiles should be avoided because they will lead~~ to ~~structures~~ that are ~~unstable toward~~ the ~~dynamical development of shape-distorting or~~ ''~~convective~~''~~-type motions. Here are a few well-known examples~~.
			You have arrived at this page from our [[H_BookTiledMenu#Axisymmetric_Equilibrium_Structures\|''Tiled Menu'']] by clicking on the chapter title that is also identified, below, by the light-blue-colored icon. You may proceed directly to that chapter by clicking (again) on that highlighted chapter icon. However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected. The common thread is the "Key Equation" titled, ''Gravitational Potential of an Axisymmetric Mass Distribution'', that appears at the top of the following set of chapter synopses.
	<br />		<br />
	<br />		<br />

Jet53man: Created page with "FORCETOC =Solving the Poisson Equation= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: s..."

2021-09-09T22:34:57Z

Created page with "__FORCETOC__  =Solving the Poisson Equation= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: s..."

New page

__FORCETOC__

=Solving the Poisson Equation=
{| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 150px; width: 150px; background-color:white;" |[[H_BookTiledMenu#Two-Dimensional_Configurations_.28Axisymmetric.29|Solving the Poisson Equation]]
|}
When constructing rotating equilibrium configurations that obey a [[SR#Time-Independent_Problems|barotropic equation of state]], keep in mind that certain physical variable profiles should be avoided because they will lead to structures that are unstable toward the dynamical development of shape-distorting or ''convective''-type motions. Here are a few well-known examples.
  
  
  
  
  

=See Also=
* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] — ''On the Dynamics of Revolving Fluids''
* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597)] — ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)
<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"</li></ol>
* Referring to our [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|accompanying discussion of Type 1 Riemann ellipsoids]], it seems that uniform rotation on cylinders is not required. What's going on?

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AxisymmetricConfigurations/SolvingPE - Revision history

Jet53man: /* Related Discussions */

Jet53man: /* Related Discussions */

Jet53man at 17:21, 10 September 2021

Jet53man at 17:11, 10 September 2021

Jet53man at 17:07, 10 September 2021

Jet53man: Created page with "__FORCETOC__ =Solving the Poisson Equation= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: s..."

Jet53man: Created page with "FORCETOC =Solving the Poisson Equation= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: s..."