User:Tohline/VE - Revision history

Tohline: /* Virial Equations (Rotating Frame) */

2021-03-28T20:33:19Z

Virial Equations (Rotating Frame)

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	[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 25, Eq. (63)		[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], § 11a, p. 25, Eq. (63) and Epilogue, p. 244, Eq. (1)
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Tohline: /* Generalization */

2018-10-28T20:15:36Z

Generalization

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	<br /> <br />		<br /> <br />
	[[User:Tohline/Appendix/References#~~Other_References~~\|Shu92]], p. 330, Eq. (24.9)		[<b>[[User:Tohline/Appendix/References#ST83\|<font color="red">ST83</font>]]</b>], p. 165, Eq. (7.1.18)<br />
			[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], p. 330, Eq. (24.9)
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Tohline: /* Generalization */

2018-04-03T23:03:11Z

Generalization

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	[ [[User:Tohline/Appendix/References#Other_References\|Shu92]], pp. 329-330 ] <font color="orange">If we were to multiply [this modified Euler equation] by <math>~x_m</math> and integrate over volume <math>~V</math>, we would get the [appropriately modified] ''tensor virial theorem'', the off-diagonal elements of which carry information concerning angular-momentum conservation (see Chandrasekhar's</font> EFE <font color="orange">book for an exposition). [Here] we shall be more interested in the trace of the tensor equation, which we may derive by simply multiplying [the modified Euler equation] by <math>~x_i</math> (with an implicit summation over repeated indices) and integrating over <math>~V</math></font>. The resulting relation governing the equilibrium of stationary states (see [[User:Tohline/Appendix/References#Other_References\|Shu92]] for derivation details), as we shall reference it, is the		[ [[User:Tohline/Appendix/References#Other_References\|Shu92]], pp. 329-330 ] <font color="orange">If we were to multiply [this modified Euler equation] by <math>~x_m</math> and integrate over volume <math>~V</math>, we would get the [appropriately modified] ''tensor virial theorem'', the off-diagonal elements of which carry information concerning angular-momentum conservation (see Chandrasekhar's</font> EFE <font color="orange">book for an exposition). [Here] we shall be more interested in the trace of the tensor equation, which we may derive by simply multiplying [the modified Euler equation] by <math>~x_i</math> (with an implicit summation over repeated indices) and integrating over <math>~V</math></font>. The resulting relation governing the equilibrium of stationary states (see [[User:Tohline/Appendix/References#Other_References\|Shu92]] for derivation details), as we shall reference it, is the
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	<span id="GenTVE"><font color="#770000">'''Generalized Scalar Virial ~~Theorm~~'''</font></span><br />		<span id="GenTVE"><font color="#770000">'''Generalized Scalar Virial Theorem'''</font></span><br />
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Tohline: /* Setting the Stage */

2017-07-06T01:45:55Z

Setting the Stage

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	{{User:Tohline/Math/EQ_Poisson01}}		{{User:Tohline/Math/EQ_Poisson01}}
	[[Image:LSU_OriginButton.jpg\|125px\|link=http://www.vistrails.org/index.php/User:Tohline/SR/PoissonOrigin]]		[[Image:LSU_OriginButton.jpg\|125px\|link=http://www.vistrails.org/index.php/User:Tohline/SR/PoissonOrigin#Origin_of_the_Poisson_Equation]]
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Tohline: /* Setting the Stage */

2017-07-05T22:33:06Z

Setting the Stage

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	{{User:Tohline/Math/EQ_Poisson01}}		{{User:Tohline/Math/EQ_Poisson01}}
			[[Image:LSU_OriginButton.jpg\|125px\|link=http://www.vistrails.org/index.php/User:Tohline/SR/PoissonOrigin]]
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Tohline: /* Derivation */

2017-06-17T16:58:35Z

Derivation

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	where,		<span id="MOItensor">where,</span>
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Tohline: /* First-Order Virial Equations */

2017-06-17T16:02:37Z

First-Order Virial Equations

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	where the moments of inertia about the three separate principal axes <math>(i = 1,2,3)</math> are defined by the expressions,		where the <span id="MomentOfInertia">moments of inertia</span> about the three separate principal axes <math>(i = 1,2,3)</math> are defined by the expressions,
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	Thus, the first-order virial equation(s) <font color="#007700">expresses the uniform motion of the center of mass of the system</font>.		Thus, the first-order virial equation(s) <font color="#007700">expresses the uniform motion of the center of mass of the system</font>.


	===Second-Order Tensor Virial Equations===		===Second-Order Tensor Virial Equations===

Tohline: /* Virial Equations (Rotating Frame) */

2016-11-18T21:22:35Z

Virial Equations (Rotating Frame)

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	[[User:Tohline/Appendix/References#~~Key_Parallel_References~~\|EFE]], p. 25, Eq. (63)		[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], p. 25, Eq. (63)
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Tohline: /* Scalar Virial Theorem */

2016-11-18T21:21:24Z

Scalar Virial Theorem

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	[[User:Tohline/Appendix/References#~~Other_References~~\|EFE]]		[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]
	Reference		Reference
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	[[User:Tohline/Appendix/References#~~Key_Parallel_References~~\|BT87]], p. 213, Eq. (4-79)		[<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-79)
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	(In a footnote to their Equation 4-79, [[User:Tohline/Appendix/References#~~Key_Parallel_References~~\|BT87]] point out that the ''scalar virial theorem'' was first proved by R. Clausius in 1870; see various links to this work under our [[User:Tohline/VE#Related_Discussions\|"Related Discussions" subsection, below]].)		(In a footnote to their Equation 4-79, [<b>[[User:Tohline/Appendix/References#BT87\|<font color="red">BT87</font>]]</b>] point out that the ''scalar virial theorem'' was first proved by R. Clausius in 1870; see various links to this work under our [[User:Tohline/VE#Related_Discussions\|"Related Discussions" subsection, below]].)

	====Generalization====		====Generalization====

Tohline: /* Steady State (Virial Equilibrium) */

2016-11-18T21:19:22Z

Steady State (Virial Equilibrium)

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	====Steady State (Virial Equilibrium)====		====Steady State (Virial Equilibrium)====
	[ [[User:Tohline/Appendix/References#~~Other_References~~\|EFE]] §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>		[<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b> §11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
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