User:Jaycall/KillingVectorApproach - Revision history

Tohline: /* Response from Jay */ Fixed sign error in eq. CV.02

2010-07-15T14:43:08Z

Response from Jay: Fixed sign error in eq. CV.02

← Older revision		Revision as of 14:43, 15 July 2010
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	<td align="right">		<td align="right">
	<math>		<math>
	\frac{d\Xi}{dt} \equiv \frac{d\ln C_2}{dt}		\frac{d\Xi}{dt} \equiv - \frac{d\ln C_2}{dt}
	</math>		</math>
	</td>		</td>
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	\biggl[ \biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2 \frac{\partial \ln h_1}{\partial\ln\lambda_2} + \frac{\partial \ln h_2}{\partial \ln\lambda_2}\biggr] \frac{d \ln{\lambda}_2}{dt}		\biggl[ \biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2 \frac{\partial \ln h_1}{\partial\ln\lambda_2} + \frac{\partial \ln h_2}{\partial \ln\lambda_2}\biggr] \frac{d \ln{\lambda}_2}{dt}
	</math>		</math>
			</td>
			</tr>

			<tr>
			<td align="left" colspan="3">
			<font color="red"><b>NOTE:</b> Sign error fixed on 15 July 2010</font>. Specifically, <math>d\Xi/dt \equiv d\ln C_2/dt</math> changed to <math>d\Xi/dt \equiv - d\ln C_2/dt</math>.
	</td>		</td>
	</tr>		</tr>

Jaycall: /* Killing Vector Approach to Integrals of Motion Problem */

2010-07-12T15:27:44Z

Killing Vector Approach to Integrals of Motion Problem

← Older revision		Revision as of 15:27, 12 July 2010
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	=~~Killing~~ Vector Approach to Integrals of Motion Problem=		=Characteristic Vector Approach to Integrals of Motion Problem=

	Motivated by our recent work in relativistic fluid dynamics, one insightful approach to the integrals of motion problem involves rewriting the equations of motion in terms of an arbitrary vector field, which we're currently calling a ''characteristic vector''. The benefit here is that we can produce a mathematical condition that tells us what combintations of variables should be grouped together in order to produce conserved quantities. The details follow.		Motivated by our recent work in relativistic fluid dynamics, one insightful approach to the integrals of motion problem involves rewriting the equations of motion in terms of an arbitrary vector field, which we're currently calling a ''characteristic vector''. The benefit here is that we can produce a mathematical condition that tells us what combintations of variables should be grouped together in order to produce conserved quantities. The details follow.

Jaycall at 15:23, 12 July 2010

2010-07-12T15:23:39Z

← Older revision		Revision as of 15:23, 12 July 2010
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	where <math>m</math> is the mass of the test particle, the <math>\lambda</math>s are the chosen coordinates, and the <math>\dot{\lambda}</math>s are the coordinate velocities (or the total time-derivatives of the coordinates), the <math>h</math>s are the metric scale factors associated with each of the coordinates, <math>\Phi</math> is the gravitational potential, and the <math>\Gamma</math>s are the Christoffel symbols.		where <math>m</math> is the mass of the test particle, the <math>\lambda</math>s are the chosen coordinates, and the <math>\dot{\lambda}</math>s are the coordinate velocities (or the total time-derivatives of the coordinates), the <math>h</math>s are the metric scale factors associated with each of the coordinates, <math>\Phi</math> is the gravitational potential, and the <math>\Gamma</math>s are the Christoffel symbols.

			==Metric Components, Orthogonality Condition, and Scale Factors==

			For any coordinate system it is possible to express the components of the metric in terms of the partial derivatives defining the transformation from any other coordinate system. Written in terms of the transformation from cylindrical coordinates, each metric component is

			<div align="center">
			<math>
			g_{ij} = \left( \partial_i R \right) \left( \partial_j R \right) + \left( \partial_i z \right) \left( \partial_j z \right) + R^2 \left( \partial_i \phi \right) \left( \partial_j \phi \right) .
			</math>
			</div>

			The orthogonality condition is that the metric be diagonal. That is, each off-diagonal component of the metric must equal zero.

			<div align="center">
			<math>
			\left( \partial_i R \right) \left( \partial_j R \right) + \left( \partial_i z \right) \left( \partial_j z \right) + R^2 \left( \partial_i \phi \right) \left( \partial_j \phi \right) = 0_{ij}, \ \ \ \ \ i \neq j .
			</math>
			</div>

			If the coordinates should be orthogonal, then scale factors can also be defined in terms of the diagonal components of the metric such that <math>g_{ii} = {h_i}^2</math>. This results in the following definition of the scale factors if the orthogonality condition is met.

			<div align="center">
			<math>
			h_i = \left[ \left( \partial_i R \right)^2 + \left( \partial_i z \right)^2 + R^2 \left( \partial_i \phi \right)^2 \right]^{1/2} .
			</math>
			</div>

	==Christoffel Symbols==		==Christoffel Symbols==

Tohline: /* Christoffel Symbols */ Correct one expression for Christoffel symbols

2010-06-10T21:16:22Z

Christoffel Symbols: Correct one expression for Christoffel symbols

← Older revision		Revision as of 21:16, 10 June 2010
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	<td align="center">		<td align="center">
	<math>		<math>
	\Gamma^i_{ij} = \Gamma^i_{ji} = \frac{\~~partial_i h_j~~}{~~h_j~~}		\Gamma^i_{ij} = \Gamma^i_{ji} = \frac{\partial_j h_i}{h_i}
	</math>		</math>
	</td>		</td>
	<td align="center" colspan="2">		<td align="center" colspan="2">
	<font color="~~red~~">~~Shouldn't this be:~~</~~font~~>~~  ~~		otherwise<br/>
	~~<math>\frac{\partial_j h_i}{h_i}~~</~~math~~> ?		<font color="blue" size="-2">
			(Mistake in original expression corrected <br/>06/10/2010, [[User_talk:Jaycall#Killing_Vector_Approach\|as discussed]].)
			</font>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Identification of Conserved Quantity */ Add equation number to first equation

2010-06-06T21:57:52Z

Identification of Conserved Quantity: Add equation number to first equation

← Older revision		Revision as of 21:57, 6 June 2010
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	The trick is to introduce an arbitrary vector field, which will be used to build a location-dependent weighted linear combination of the equations of motion. Since the equations of motion associated with each of the coordinates collectively form a single vector equation, we will just form an inner product of our characteristic vector with each side of the vector equation of motion. The result is		The trick is to introduce an arbitrary vector field, which will be used to build a location-dependent weighted linear combination of the equations of motion. Since the equations of motion associated with each of the coordinates collectively form a single vector equation, we will just form an inner product of our characteristic vector with each side of the vector equation of motion. The result is

			<span id="CV.01"><table align="right" border="1" cellpadding="10" width="10%">
			<tr><th><font color="darkblue">CV.01</font></th></tr>
			</table></span>
	<div align="center">		<div align="center">
	<math>		<math>
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	So we're looking for a vector <math>\vec{C}</math> such that		So we're looking for a vector <math>\vec{C}</math> such that

	~~<span id="CV.01"><table align="right" border="1" cellpadding="10" width="10%">~~
	~~<tr><th><font color="darkblue">CV.01</font></th></tr>~~
	~~</table></span>~~
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* Strategic Choice of Characteristic Vector */ Add a couple of right-justified equation numbers

2010-06-06T21:25:55Z

Strategic Choice of Characteristic Vector: Add a couple of right-justified equation numbers

← Older revision		Revision as of 21:25, 6 June 2010
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	So we're looking for a vector <math>\vec{C}</math> such that		So we're looking for a vector <math>\vec{C}</math> such that

			<span id="CV.01"><table align="right" border="1" cellpadding="10" width="10%">
			<tr><th><font color="darkblue">CV.01</font></th></tr>
			</table></span>
	<div align="center">		<div align="center">
	<math>		<math>
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	And limiting our interest to motion within the meridional plane (setting <math>\dot{\lambda}_3 = 0</math>) and simplifying		And limiting our interest to motion within the meridional plane (setting <math>\dot{\lambda}_3 = 0</math>) and simplifying

			<span id="CV.02"><table align="right" border="1" cellpadding="10" width="10%">
			<tr><th><font color="darkblue">CV.02</font></th></tr>
			</table></span>
	<table align="center" border="1" cellpadding="5">		<table align="center" border="1" cellpadding="5">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>		<math>
	\frac{d\Xi}{dt}		\frac{d\Xi}{dt} \equiv \frac{d\ln C_2}{dt}
	</math>		</math>
	</td>		</td>

Tohline: Move subsection "application to T3 coordinates" to entirely new page

2010-06-06T21:12:48Z

Move subsection "application to T3 coordinates" to entirely new page

Show changes

Tohline: /* Christoffel Symbols */ Joel's question regarding tabulated expression

2010-06-06T15:34:13Z

Christoffel Symbols: Joel's question regarding tabulated expression

← Older revision		Revision as of 15:34, 6 June 2010
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	</td>		</td>
	<td align="center" colspan="2">		<td align="center" colspan="2">
	~~otherwise~~		<font color="red">Shouldn't this be:</font>
			<math>\frac{\partial_j h_i}{h_i}</math> ?
	</td>		</td>
	</tr>		</tr>

Tohline: /* Application to T3 Coordinates */ Expand discussion of application to T3 coordinates

2010-06-06T01:51:40Z

Application to T3 Coordinates: Expand discussion of application to T3 coordinates

← Older revision		Revision as of 01:51, 6 June 2010
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	<table align="center" border="1" cellpadding="5">		<table align="center" border="1" cellpadding="5">
	<tr>		<tr>
			<td align="right">

			</td>
	<td align="right">		<td align="right">
	<math>		<math>
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	<math>		<math>
	\biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2 \frac{\partial \ln h_1}{\partial\ln\lambda_2} + \frac{\partial \ln h_2}{\partial \ln\lambda_2}		\biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2 \frac{\partial \ln h_1}{\partial\ln\lambda_2} + \frac{\partial \ln h_2}{\partial \ln\lambda_2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2 \biggl( \frac{q h_1 h_2 \lambda_2}{\lambda_1 } \biggr)^2 - ( qh_1^2 )^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ (h_1 \dot{\lambda}_1)^2 ( q h_1 h_2 \lambda_2 )^2 - (h_2 \dot{\lambda}_2)^{2} ( qh_1^2 \lambda_1 )^2 \biggr](h_2 \lambda_1 \dot{\lambda}_2)^{-2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \biggl(\frac{\dot{\lambda}_1}{\lambda_1}\biggr)^2 - \biggl( \frac{\dot{\lambda}_2}{\lambda_2} \biggr)^2 \biggr]( q h_1^2 h_2 \lambda_1 \lambda_2 )^2 (h_2 \lambda_1 \dot{\lambda}_2)^{-2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2} \biggr] \biggl[ \frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2} \biggr] \biggl( \frac{ q h_1^2 \lambda_2}{\dot{\lambda}_2} \biggr)^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\Rightarrow</math>
			</td>
			<td align="right">
			<math>
			\frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2} \biggr] \biggl[ \frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2} \biggr] ( q h_1^2 )^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2} \biggr] \frac{d\ln h_2}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{\ln(\lambda_1 \lambda_2)}{dt} \biggr] \frac{d\ln h_2}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			or
			</td>
			<td align="right">
			<math>
			\frac{\dot{C}_2}{C_2}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2} \biggr] \frac{d\ln h_2}{dt} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="center">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl[ \frac{\dot{\lambda}_1}{\lambda_1}\biggr] \frac{d\ln h_2}{dt} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1} + \frac{d\ln h_2}{dt}
	</math>		</math>
	</td>		</td>

Tohline: Begin application of characteristic vector formalism to T3 Coordinates

2010-06-06T01:04:42Z

Begin application of characteristic vector formalism to T3 Coordinates

← Older revision		Revision as of 01:04, 6 June 2010
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			{{LSU_HBook_header}}

	=Killing Vector Approach to Integrals of Motion Problem=		=Killing Vector Approach to Integrals of Motion Problem=

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	</tr>		</tr>
	</table>		</table>

			=Application to T3 Coordinates=

			On an [[User:Tohline/Appendix/Ramblings/T3Integrals\|accompanying page]] we have defined, and detailed many properties of, a ''T3 Coordinate System''. Here we explore how the differential equation derived above that defines our desired characteristic vector behaves in the specific case of T3 Coordinates.

			<table align="center" border="1" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			\frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1}
			</math>
			</td>
			<td align="center">
			<math>
			=
			</math>
			</td>
			<td align="left">
			<math>
			\biggl(\frac{h_1 \dot{\lambda}_1}{h_2 \dot{\lambda}_2}\biggr)^2 \frac{\partial \ln h_1}{\partial\ln\lambda_2} + \frac{\partial \ln h_2}{\partial \ln\lambda_2}
			</math>
			</td>
			</tr>

			</table>



			<br />
			{{LSU_HBook_footer}}