User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase - Revision history

Tohline: /* Another Thought */ More discussion

2010-06-21T21:27:49Z

Another Thought: More discussion

← Older revision		Revision as of 21:27, 21 June 2010
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	</div>		</div>

			Hence, [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase#T3Q.01\|Equation T3Q.01]] can be rewritten as,
			<div align="center">
			<math>
			\frac{d}{dt}\biggl( \frac{\dot{\lambda_2}}{\lambda_2} \biggr) + \biggl[\frac{d\ln \lambda_2}{dt} \biggr]^2 = \biggl[\frac{d\ln (\Lambda-1)^{1/2}}{dt} \biggr]^2
			</math><br /><br />

			<math>
			\Rightarrow ~~~~~ \frac{d}{dt}\biggl( \frac{d\ln\lambda_2}{dt} \biggr) = \biggl[\frac{d\ln (\Lambda-1)^{1/2}}{dt} \biggr]^2 - \biggl[\frac{d\ln \lambda_2}{dt} \biggr]^2 = \biggl[\frac{d\ln (\Lambda-1)^{1/2}}{dt} + \frac{d\ln \lambda_2}{dt} \biggr] \biggl[\frac{d\ln (\Lambda-1)^{1/2}}{dt} - \frac{d\ln \lambda_2}{dt} \biggr]
			</math><br /><br />

			<math>
			\Rightarrow ~~~~~ \frac{d}{dt}\biggl( \frac{d\ln\lambda_2}{dt} \biggr) = \frac{d\ln [(\Lambda-1)^{1/2}\lambda_2]}{dt} \cdot \frac{d\ln [(\Lambda-1)^{1/2} \lambda_2^{-1}]}{dt}
			</math>
			</div>
			Can we gain any insight from this form of the <math>2^\mathrm{nd}</math> component of the equation of motion? For example, can the convert the RHS into the total time-derivative of some function?

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Tohline: /* Special Case (Quadratic) */ Begin another thought

2010-06-21T21:06:45Z

Special Case (Quadratic): Begin another thought

← Older revision		Revision as of 21:06, 21 June 2010
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	But can we learn something valuable from this failed separation of variables approach???		But can we learn something valuable from this failed separation of variables approach???

			===Another Thought===

			It can easily be shown that, in general,
			<div align="center">
			<math>
			\frac{d}{dt}\biggl( \frac{\dot{F}}{F} \biggr) = \frac{\ddot{F}}{F} - \biggl[\frac{d\ln F}{dt} \biggr]^2 .
			</math>
			</div>


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Tohline: /* Solution Strategy */ modify wording slightly

2010-06-12T15:11:32Z

Solution Strategy: modify wording slightly

← Older revision		Revision as of 15:11, 12 June 2010
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	</math>		</math>
	</div>		</div>
	This would be a fantastically simple result, if it proved to be a proper solution to the governing equation of motion. Unfortunately, if this relatively elementary equation is differentiated in an effort to reproduce [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase#T3Q.01\|'''Equation T3Q.01''']], we find that an additional undesirable term appears that involves the second derivative of a function containing the variable <math>\Lambda</math>. It can be shown that this undesirable term goes to zero if <math>\sqrt{n}</math> is assumed to be independent of time (as we did indeed assume, above). Unfortunately, in reality, this does not seem to be a desirable assumption for the physical problem in which we have interest, so we must conclude that the derived elementary equation is not the desired solution of the equation of motion.		This would be a fantastically simple result, if it proved to be a proper solution to the governing equation of motion. Unfortunately, if this relatively elementary equation is differentiated twice in an effort to reproduce [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase#T3Q.01\|'''Equation T3Q.01''']], we find that an additional undesirable term appears that involves the second derivative of a function containing the variable <math>\Lambda</math>. It can be shown that this undesirable term goes to zero if <math>\sqrt{n}</math> is assumed to be independent of time (as we did indeed assume, above). Unfortunately, in reality, this does not seem to be a desirable assumption for the physical problem in which we have interest, so we must conclude that the derived elementary equation is not the desired solution of the equation of motion.

	But can we learn something valuable from this failed separation of variables approach???		But can we learn something valuable from this failed separation of variables approach???

Tohline: /* Equation of Motion */ Add slightly different form of one expression

2010-06-12T15:05:38Z

Equation of Motion: Add slightly different form of one expression

← Older revision		Revision as of 15:05, 12 June 2010
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	<br /><br />		<br /><br />
	<math>		<math>
	\Rightarrow ~~~~~ \frac{\ddot{\lambda}_2}{\lambda_2} = \biggl[\frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2} \biggr]\frac{d\ln h_2 }{dt} .		\Rightarrow ~~~~~ \frac{\ddot{\lambda}_2}{\lambda_2} = \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_1/\lambda_2)}{dt} \biggr]\frac{d\ln h_2 }{dt} .
	</math>		</math>
	</div>		</div>

Tohline: /* Solution Strategy */ Proceed with separation of variables approach

2010-06-12T01:22:11Z

Solution Strategy: Proceed with separation of variables approach

← Older revision		Revision as of 01:22, 12 June 2010
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	where, <math>\Lambda_0</math> is given by the coordinate ratio at time <math>t=0</math>, specifically, <math>\Lambda_0 \equiv \sqrt{1 + (2\lambda_1^0/\lambda_2^0)^2}</math>.		where, <math>\Lambda_0</math> is given by the coordinate ratio at time <math>t=0</math>, specifically, <math>\Lambda_0 \equiv \sqrt{1 + (2\lambda_1^0/\lambda_2^0)^2}</math>.

	~~Minor addition~~ to ~~check wiki saving capabilities!~~		Now, if we replace "<math>\sqrt{n}~t</math>" in the first of these expressions by the second expression, we find,
			<div align="center">
			<math>
			\frac{\lambda_2}{\lambda^0_2} = \exp{\biggl[ -\frac{1}{\sqrt{2}} \ln\biggl( \frac{\Lambda-1}{\Lambda_0 -1} \biggr) \biggr]} = \biggl( \frac{\Lambda_0 -1}{\Lambda-1} \biggr)^{1/\sqrt{2}} .
			</math>
			</div>
			This would be a fantastically simple result, if it proved to be a proper solution to the governing equation of motion. Unfortunately, if this relatively elementary equation is differentiated in an effort to reproduce [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase#T3Q.01\|'''Equation T3Q.01''']], we find that an additional undesirable term appears that involves the second derivative of a function containing the variable <math>\Lambda</math>. It can be shown that this undesirable term goes to zero if <math>\sqrt{n}</math> is assumed to be independent of time (as we did indeed assume, above). Unfortunately, in reality, this does not seem to be a desirable assumption for the physical problem in which we have interest, so we must conclude that the derived elementary equation is not the desired solution of the equation of motion.

			But can we learn something valuable from this failed separation of variables approach???


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Tohline: Minor additional sentence to force a new wiki save

2010-06-10T20:22:24Z

Minor additional sentence to force a new wiki save

← Older revision		Revision as of 20:22, 10 June 2010
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	where, <math>\Lambda_0</math> is given by the coordinate ratio at time <math>t=0</math>, specifically, <math>\Lambda_0 \equiv \sqrt{1 + (2\lambda_1^0/\lambda_2^0)^2}</math>.		where, <math>\Lambda_0</math> is given by the coordinate ratio at time <math>t=0</math>, specifically, <math>\Lambda_0 \equiv \sqrt{1 + (2\lambda_1^0/\lambda_2^0)^2}</math>.

			Minor addition to check wiki saving capabilities!

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Tohline: /* Solution Strategy */ More on solution strategy

2010-06-10T19:43:17Z

Solution Strategy: More on solution strategy

← Older revision		Revision as of 19:43, 10 June 2010
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	<div align="center">		<div align="center">
	<math>		<math>
	\ddot{\lambda}_2 = \frac{n}{2}\lambda_2		\ddot{\lambda}_2 = \frac{n}{2}\lambda_2 ;
	</math>		</math>
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	\frac{d\ln(\Lambda-1)}{dt} = \sqrt{n}		\frac{d\ln(\Lambda-1)}{dt} = \sqrt{n} .
	</math>		</math>
	</div>		</div>
			Now I suppose that, in general, <math>n</math> should be allowed to vary with time, but for exploratory purposes, let's assume that <math>n</math> is a constant. The solution to the LHS's <math>2^\mathrm{nd}</math>-order ODE is,
			<div align="center">
			<math>
			\lambda_2 = \lambda^0_2 \exp{[-\sqrt{n/2}~t]} ,
			</math>
			</div>
			where, <math>\lambda^0_2</math> is the coordinate position <math>\lambda_2</math> at time <math>t=0</math>. The solution to the RHS's <math>1^\mathrm{st}</math>-order ODE is,
			<div align="center">
			<math>
			\sqrt{n} t = \ln\biggl( \frac{\Lambda-1}{\Lambda_0 -1} \biggr) ,
			</math>
			</div>
			where, <math>\Lambda_0</math> is given by the coordinate ratio at time <math>t=0</math>, specifically, <math>\Lambda_0 \equiv \sqrt{1 + (2\lambda_1^0/\lambda_2^0)^2}</math>.



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Tohline: /* Special Case (Quadratic) */ Begin discussion of solution strategy

2010-06-10T19:17:06Z

Special Case (Quadratic): Begin discussion of solution strategy

← Older revision		Revision as of 19:17, 10 June 2010
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	</math><br /><br />		</math><br /><br />
	</div>		</div>


	<span id="T3Q.01"><table align="right" border="1" cellpadding="10" width="10%">		<span id="T3Q.01"><table align="right" border="1" cellpadding="10" width="10%">
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	</div>		</div>

			===Solution Strategy===
			I'm not sure whether the following strategy is fully legitimate, but let's explore it anyway. Because the LHS of [[User:Tohline/Appendix/Ramblings/T3Integrals/QuadraticCase#T3Q.01\|'''Equation T3Q.01''']] displays an explicit dependence only on the coordinate <math>\lambda_2</math> while the RHS displays an explicit dependence only on <math>\Lambda</math> — that is, only on the ''ratio'' of the two coordinates <math>\lambda_1/\lambda_2</math> — perhaps we can use a ''separation of variables'' technique to derive a solution. Specifically, suppose the LHS and the RHS separately are set equal to the same value, call it <math>n</math>.

			Then, for the LHS:
			<div align="center">
			<math>
			\ddot{\lambda}_2 = \frac{n}{2}\lambda_2
			</math>
			</div>

			And, for the RHS:
			<div align="center">
			<math>
			\frac{d\ln(\Lambda-1)}{dt} = \sqrt{n}
			</math>
			</div>


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Tohline: /* Equation of Motion */ Further elaboration on EOM

2010-06-10T19:01:22Z

Equation of Motion: Further elaboration on EOM

← Older revision		Revision as of 19:01, 10 June 2010
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	<math>		<math>
	\Rightarrow ~~~~~ \frac{\ddot{\lambda}_2}{\lambda_2} = \biggl[\frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2} \biggr]\frac{d\ln h_2 }{dt} .		\Rightarrow ~~~~~ \frac{\ddot{\lambda}_2}{\lambda_2} = \biggl[\frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2} \biggr]\frac{d\ln h_2 }{dt} .
			</math>
			</div>
			Based on the derivations provided above, both factors that make up the term on the RHS of this expression can be written entirely in terms of the variable, <math>\Lambda</math>. This allows us to rewrite [[User:Tohline/Appendix/Ramblings/T3Integrals#EOM.01\|'''Equation EOM.01''']] as,
			<div align="center">
			<math>
			\frac{\ddot{\lambda}_2}{\lambda_2} = \biggl[\frac{\Lambda}{\Lambda^2-1} \frac{d\Lambda}{dt}\biggr]\biggl[ \frac{1}{2\Lambda} \biggl( \frac{\Lambda + 1}{\Lambda - 1} \biggr) \frac{d\Lambda}{dt} \biggr] = \frac{1}{2(\Lambda - 1)^2} \biggl[\frac{d\Lambda}{dt} \biggr]^2
			</math><br /><br />
			</div>


			<span id="T3Q.01"><table align="right" border="1" cellpadding="10" width="10%">
			<tr><th><font color="darkblue">T3Q.01</font></th></tr>
			</table></span>
			<div align="center">
			<math>
			\Rightarrow ~~~~~ 2\frac{\ddot{\lambda}_2}{\lambda_2} = \biggl[\frac{d\ln(\Lambda-1)}{dt} \biggr]^2.
	</math>		</math>
	</div>		</div>

Tohline: /* Special Case (Quadratic) */ Move on to the equation of motion in this special case

2010-06-10T14:05:34Z

Special Case (Quadratic): Move on to the equation of motion in this special case

← Older revision		Revision as of 14:05, 10 June 2010
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{1}{2} \biggl( \frac{\Lambda +1}{\Lambda-1} \biggr) \frac{d\ln\Lambda}{dt}		\frac{1}{2} \biggl( \frac{\Lambda +1}{\Lambda-1} \biggr) \frac{d\ln\Lambda}{dt} .
	</math>		</math>
	</td>		</td>
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	<math>		<math>
	(h_1 \lambda_1)^2 \frac{d\ln h_1}{dt} + (h_2 \lambda_2)^2 \frac{d\ln h_2}{dt} = 0 .		(h_1 \lambda_1)^2 \frac{d\ln h_1}{dt} + (h_2 \lambda_2)^2 \frac{d\ln h_2}{dt} = 0 .
			</math>
			</div>

			===Equation of Motion===

			According to [[User:Tohline/Appendix/Ramblings/T3Integrals#EOM.01\|'''Equation EOM.01''']], as derived elsewhere in the context of T3 coordinates, a general expression for the second component of the equation of motion is,
			<div align="center">
			<math>
			\frac{d(h_2 \dot{\lambda}_2)}{dt} = \biggl(\frac{\lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr) \frac{dh_2}{dt}
			</math>
			<br /><br />
			<math>
			\Rightarrow ~~~~~ \frac{\ddot{\lambda}_2}{\lambda_2} = \biggl[\frac{\dot{\lambda}_1}{\lambda_1} - \frac{\dot{\lambda}_2}{\lambda_2} \biggr]\frac{d\ln h_2 }{dt} .
	</math>		</math>
	</div>		</div>