User talk:Jaycall - Revision history

Tohline: /* Two Views of Equation of Motion */ Quick response

2010-07-15T22:28:36Z

Two Views of Equation of Motion: Quick response

← Older revision		Revision as of 22:28, 15 July 2010
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	::Joel, I'm sorry yesterday's excitement didn't pan out. It would have been really neat. Well, at least I learned something. On my drive home yesterday I was thinking about what it meant for there to be a constraint that involved only the coordinates (and not their rates of change). We were uncomfortable with this idea, but now I understand why in general no conserved quantity can take this form. If there were a constraint on only the coordinates, then this would limit a particle to a curve or a surface...a subspace of physical space (as opposed to a subspace of phase space). Yet it would definitely be possible to get a particle to leave the subspace by giving it an initial velocity in any direction normal to the subspace. Any conserved quantity must definitely involve the coordinate velocities (unless you're artificially restricting the motion). --[[User:Jaycall\|Jaycall]] 14:15, 15 July 2010 (MDT)		::Joel, I'm sorry yesterday's excitement didn't pan out. It would have been really neat. Well, at least I learned something. On my drive home yesterday I was thinking about what it meant for there to be a constraint that involved only the coordinates (and not their rates of change). We were uncomfortable with this idea, but now I understand why in general no conserved quantity can take this form. If there were a constraint on only the coordinates, then this would limit a particle to a curve or a surface...a subspace of physical space (as opposed to a subspace of phase space). Yet it would definitely be possible to get a particle to leave the subspace by giving it an initial velocity in any direction normal to the subspace. Any conserved quantity must definitely involve the coordinate velocities (unless you're artificially restricting the motion). --[[User:Jaycall\|Jaycall]] 14:15, 15 July 2010 (MDT)

			:::Good observation. --[[User:Tohline\|Tohline]] 16:28, 15 July 2010 (MDT)

	==Special Quadratic Case==		==Special Quadratic Case==

Jaycall: /* Two Views of Equation of Motion */

2010-07-15T20:16:02Z

Two Views of Equation of Motion

← Older revision		Revision as of 20:16, 15 July 2010
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	:Jay, very good catch! Looking back through the history of that wiki page, it is clear that I am the one who introduced this sign error. I mistakingly set <math>\Xi = \ln C_2</math> in equation [[User:Jaycall/KillingVectorApproach#CV.02\|CV.02]] at the same time that I "Added a couple of right-justified equation numbers." I'll march through the relevent wiki pages today and attempt to correct this mistake wherever it appears. --[[User:Tohline\|Tohline]] 08:30, 15 July 2010 (MDT)		:Jay, very good catch! Looking back through the history of that wiki page, it is clear that I am the one who introduced this sign error. I mistakingly set <math>\Xi = \ln C_2</math> in equation [[User:Jaycall/KillingVectorApproach#CV.02\|CV.02]] at the same time that I "Added a couple of right-justified equation numbers." I'll march through the relevent wiki pages today and attempt to correct this mistake wherever it appears. --[[User:Tohline\|Tohline]] 08:30, 15 July 2010 (MDT)

			::Joel, I'm sorry yesterday's excitement didn't pan out. It would have been really neat. Well, at least I learned something. On my drive home yesterday I was thinking about what it meant for there to be a constraint that involved only the coordinates (and not their rates of change). We were uncomfortable with this idea, but now I understand why in general no conserved quantity can take this form. If there were a constraint on only the coordinates, then this would limit a particle to a curve or a surface...a subspace of physical space (as opposed to a subspace of phase space). Yet it would definitely be possible to get a particle to leave the subspace by giving it an initial velocity in any direction normal to the subspace. Any conserved quantity must definitely involve the coordinate velocities (unless you're artificially restricting the motion). --[[User:Jaycall\|Jaycall]] 14:15, 15 July 2010 (MDT)

	==Special Quadratic Case==		==Special Quadratic Case==

Tohline: /* Two Views of Equation of Motion */ got rid of time-derivative in relating \Xi to lnC_2

2010-07-15T14:42:40Z

Two Views of Equation of Motion: got rid of time-derivative in relating \Xi to lnC_2

← Older revision		Revision as of 14:42, 15 July 2010
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	Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all; your intuition was right. --[[User:Jaycall\|Jaycall]] 18:50, 14 July 2010 (MDT)		Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all; your intuition was right. --[[User:Jaycall\|Jaycall]] 18:50, 14 July 2010 (MDT)

	:Jay, very good catch! Looking back through the history of that wiki page, it is clear that I am the one who introduced this sign error. I mistakingly set <math>\Xi = d\ln C_2~~/dt~~</math> in equation [[User:Jaycall/KillingVectorApproach#CV.02\|CV.02]] at the same time that I "Added a couple of right-justified equation numbers." I'll march through the relevent wiki pages today and attempt to correct this mistake wherever it appears. --[[User:Tohline\|Tohline]] 08:30, 15 July 2010 (MDT)		:Jay, very good catch! Looking back through the history of that wiki page, it is clear that I am the one who introduced this sign error. I mistakingly set <math>\Xi = \ln C_2</math> in equation [[User:Jaycall/KillingVectorApproach#CV.02\|CV.02]] at the same time that I "Added a couple of right-justified equation numbers." I'll march through the relevent wiki pages today and attempt to correct this mistake wherever it appears. --[[User:Tohline\|Tohline]] 08:30, 15 July 2010 (MDT)

	==Special Quadratic Case==		==Special Quadratic Case==

Tohline: /* Two Views of Equation of Motion */ Joel confesses introducing sign error :)

2010-07-15T14:30:07Z

Two Views of Equation of Motion: Joel confesses introducing sign error :)

← Older revision		Revision as of 14:30, 15 July 2010
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	Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all; your intuition was right. --[[User:Jaycall\|Jaycall]] 18:50, 14 July 2010 (MDT)		Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all; your intuition was right. --[[User:Jaycall\|Jaycall]] 18:50, 14 July 2010 (MDT)

			:Jay, very good catch! Looking back through the history of that wiki page, it is clear that I am the one who introduced this sign error. I mistakingly set <math>\Xi = d\ln C_2/dt</math> in equation [[User:Jaycall/KillingVectorApproach#CV.02\|CV.02]] at the same time that I "Added a couple of right-justified equation numbers." I'll march through the relevent wiki pages today and attempt to correct this mistake wherever it appears. --[[User:Tohline\|Tohline]] 08:30, 15 July 2010 (MDT)

	==Special Quadratic Case==		==Special Quadratic Case==

Jaycall: /* Two Views of Equation of Motion */

2010-07-15T00:55:36Z

Two Views of Equation of Motion

← Older revision		Revision as of 00:55, 15 July 2010
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	::::Jay: For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines\|Direction Cosines]].--[[User:Tohline\|Tohline]] 18:47, 3 July 2010 (MDT)		::::Jay: For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines\|Direction Cosines]].--[[User:Tohline\|Tohline]] 18:47, 3 July 2010 (MDT)

	Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all. --[[User:Jaycall\|Jaycall]] 18:50, 14 July 2010 (MDT)		Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all; your intuition was right. --[[User:Jaycall\|Jaycall]] 18:50, 14 July 2010 (MDT)

	==Special Quadratic Case==		==Special Quadratic Case==

Jaycall: /* Two Views of Equation of Motion */

2010-07-15T00:50:48Z

Two Views of Equation of Motion

← Older revision		Revision as of 00:50, 15 July 2010
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	::::Jay: For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines\|Direction Cosines]].--[[User:Tohline\|Tohline]] 18:47, 3 July 2010 (MDT)		::::Jay: For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines\|Direction Cosines]].--[[User:Tohline\|Tohline]] 18:47, 3 July 2010 (MDT)

	Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all.		Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all. --[[User:Jaycall\|Jaycall]] 18:50, 14 July 2010 (MDT)

	==Special Quadratic Case==		==Special Quadratic Case==

Jaycall: /* Two Views of Equation of Motion */

2010-07-15T00:47:03Z

Two Views of Equation of Motion

← Older revision		Revision as of 00:47, 15 July 2010
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	::::Jay: For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines\|Direction Cosines]].--[[User:Tohline\|Tohline]] 18:47, 3 July 2010 (MDT)		::::Jay: For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines\|Direction Cosines]].--[[User:Tohline\|Tohline]] 18:47, 3 July 2010 (MDT)

			Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>. There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach\|characteristic vector approach]]. My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>. Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>. So in fact you ''did'' find a conserved quantity--<math>1</math>! : ) I guess it was a tautology after all.

	==Special Quadratic Case==		==Special Quadratic Case==

Jaycall: /* Special Quadratic Case */

2010-07-10T20:58:07Z

Special Quadratic Case

← Older revision		Revision as of 20:58, 10 July 2010
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	</div>		</div>
	By the way, if you don't agree that <math>\sqrt{1+(2\lambda_1/\lambda_2)^2}</math> equals <math>\cosh(2\Zeta)</math>, please let me know!--[[User:Tohline\|Tohline]] 19:49, 11 June 2010 (MDT)		By the way, if you don't agree that <math>\sqrt{1+(2\lambda_1/\lambda_2)^2}</math> equals <math>\cosh(2\Zeta)</math>, please let me know!--[[User:Tohline\|Tohline]] 19:49, 11 June 2010 (MDT)

			:: Joel: I have checked and agree that in the <math>q^2=2</math> case only, the above relation among <math>\Lambda</math>, <math>Z</math> and the T3 coordinates is correct. --[[User:Jaycall\|Jaycall]] 14:58, 10 July 2010 (MDT)

Jaycall: /* Logarithmic Derivatives of T3 Scale Factors */

2010-07-10T20:29:41Z

Logarithmic Derivatives of T3 Scale Factors

← Older revision		Revision as of 20:29, 10 July 2010
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	'''<font color="red">Mistake!</font>''' Jay: I have found a mistake in my original derivation of the logarithmic derivatives of the <math>h_1</math> scale factor. Perhaps now they will match yours. Check it out and let me know! --[[User:Tohline\|Tohline]] 12:41, 4 June 2010 (MDT)		'''<font color="red">Mistake!</font>''' Jay: I have found a mistake in my original derivation of the logarithmic derivatives of the <math>h_1</math> scale factor. Perhaps now they will match yours. Check it out and let me know! --[[User:Tohline\|Tohline]] 12:41, 4 June 2010 (MDT)

			::Joel: I am finally able to confirm your derivation of the logarithmic derivatives of the scale factors. I came at them from a different angle with a fresh head, and I am now confident that what you have is correct. Next I will look into the general case. --[[User:Jaycall\|Jaycall]] 14:29, 10 July 2010 (MDT)

	==Two Views of Equation of Motion==		==Two Views of Equation of Motion==

Tohline: /* Two Views of Equation of Motion */ Link to discussion on Direction Cosines

2010-07-04T00:47:41Z

Two Views of Equation of Motion: Link to discussion on Direction Cosines

← Older revision		Revision as of 00:47, 4 July 2010
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	:::Yes, it does. That's very helpful. --[[User:Jaycall\|Jaycall]] 09:49, 11 June 2010 (MDT)		:::Yes, it does. That's very helpful. --[[User:Jaycall\|Jaycall]] 09:49, 11 June 2010 (MDT)

			::::Jay: For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines\|Direction Cosines]].--[[User:Tohline\|Tohline]] 18:47, 3 July 2010 (MDT)

	==Special Quadratic Case==		==Special Quadratic Case==