User:Tohline/Appendix/Ramblings/T3CharacteristicVector - Revision history

Tohline: /* Implications */ Fix sign error

2010-07-15T15:18:47Z

Implications: Fix sign error

← Older revision		Revision as of 15:18, 15 July 2010
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	</table>		</table>


			<table border="1" cellpadding="10" width="90%" align="center">
			<tr>
			<td align="left">
			<font color="red"><b>NOTE:</b>The following few ''boxed-in'' sentences/expressions are incorrect.</font> They originally appeared in this discussion due to confusion that arose in conjunction with a sign error in the expression for <math>d\ln C_2/dt</math> (see the top of this page). On 14 July 2010, following a lengthy discussion between Joel and Jay of the statements colored in green, Jay spotted the sign error. (See his [[User_talk:Jaycall#Two_Views_of_Equation_of_Motion\|14 July 2010 talk-page comment]].) The proper derivation/conclusion resulting from the corrected sign error follows these ''boxed-in'' sentences/expressions.
			</td>
			</tr>
			<tr><td>
	Comparing this with the ''brute force'' derivation of the condition derived above for the characteristic vector, <math>C_2</math>, we see that the two expressions are the same if we set,		Comparing this with the ''brute force'' derivation of the condition derived above for the characteristic vector, <math>C_2</math>, we see that the two expressions are the same if we set,
	<div align="center">		<div align="center">
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	</font></b>		</font></b>

	Temp note: Joel, I don't quite understand this. Next time we get together, can you explain this page to me?		Temp note (from Jay): Joel, I don't quite understand this. Next time we get together, can you explain this page to me?

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


			Comparing this last differential equation with the ''brute force'' derivation of the condition derived above for the characteristic vector, <math>C_2</math>, we see that the two expressions are the same if we set,
			<div align="center">
			<math>
			C_2 = ( h_2^2 \dot{\lambda}_2 )^{-1} .
			</math>
			</div>

			<b><font color="darkgreen">
			At first sight, this seems to imply that we have discovered a conserved quantity. But, alas, the result is a trivial one: The resulting conserved quantity is, <math>C_2(h_2^2 \dot{\lambda}_2) = 1</math>.
			</font></b>

	==Conserved Quantity==		==Conserved Quantity==

Tohline: /* Brute Force Manipulations */ Fix sign error

2010-07-15T14:47:05Z

Brute Force Manipulations: Fix sign error

← Older revision		Revision as of 14:47, 15 July 2010
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	==Brute Force Manipulations==		==Brute Force Manipulations==

	Starting from '''[[User:Jaycall/KillingVectorApproach#CV.02\|Equation CV.02]]''', and plugging in expressions for various [[User:Tohline/Appendix/Ramblings/T3Integrals#Logarithmic_Derivatives_of_Scale_Factors\|logarithmic derivatives of the T3 scale factors]], we obtain,		Starting from '''[[User:Jaycall/KillingVectorApproach#CV.02\|Equation CV.02]]''', and plugging in expressions for various [[User:Tohline/Appendix/Ramblings/T3Integrals#Logarithmic_Derivatives_of_Scale_Factors\|logarithmic derivatives of the T3 scale factors]], we obtain [<font color="red">Note: Sign error from equation CV.02 fixed here on 15 July 2010</font>],

	<table align="center" border="0" cellpadding="5">		<table align="center" border="0" cellpadding="5">
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	<td align="right">		<td align="right">
	<math>		<math>
	\frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1}		- \frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)^{-1}
	</math>		</math>
	</td>		</td>
Line 112:		Line 112:
	<td align="right">		<td align="right">
	<math>		<math>
	\frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)		- \frac{\dot{C}_2}{C_2} \biggl(\frac{d \ln{\lambda}_2}{dt}\biggr)
	</math>		</math>
	</td>		</td>

Jaycall: Thoughts on integration

2010-07-14T17:14:54Z

Thoughts on integration

← Older revision		Revision as of 17:14, 14 July 2010
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	Although this is not all that useful in an analytic sense until we can integrate it, I wonder if it can be a guide to building a more accurate numerical model. Certainly this function can be integrated numerically, and that's got to be useful somehow...		Although this is not all that useful in an analytic sense until we can integrate it, I wonder if it can be a guide to building a more accurate numerical model. Certainly this function can be integrated numerically, and that's got to be useful somehow...

			==Thoughts on Integrating This Conserved Quantity==

			The quantity appearing inside the parentheses has an interesting symmetry. Each variable appearing without a dot in the first term appears in the same place with a dot in the second term, and vice versa. Certainly there must be some differentiation rule that will allow us to express this quantity as a total time derivative.

			On the other hand, the factor of <math>\dot{\lambda_2}</math> appearing in the denominator of the first term is troublesome. I can't think of any differentiation rule that puts a derivative in the denominator. Product rule, quotient rule, and chain rule all end up ''multiplying'' by derivatives. So I wonder if there's some way to eliminate the <math>\dot{\lambda_2}</math> in favor of undotted variables. This would require transforming the equation of motion for the <math>\dot{\lambda_2}</math> coordinate into a first-order equation. Right now, the second-order equation reads

			<div align="center">
			<math>
			\ddot{\lambda_2} + \frac{\dot{h_2}}{h_2} \dot{\lambda_2} - \frac{\dot{\lambda_1} \dot{h_2}}{\lambda_1 h_2} \lambda_2 = 0 .
			</math>
			</div>

			The first step in reducing this to a first-order equation is to perform a transformation of variables that eliminates that <math>\dot{\lambda_2}</math> term. I have successfully accomplished this. By defining <math>b \equiv {h_2}^{1/2} \lambda_2</math>, the equation can be written:

			<div align="center">
			<math>
			\ddot{b} + \left( \tfrac{1}{4} \frac{{\dot{h_2}}^2}{h_2} - \tfrac{1}{2} \frac{\ddot{h_2}}{h_2} - \frac{\dot{\lambda_1} \dot{h_2}}{\lambda_1 h_2} \right) b = 0 .
			</math>
			</div>

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Jaycall: Added section on conserved quantity specific to T3 coordinates

2010-07-12T16:11:02Z

Added section on conserved quantity specific to T3 coordinates

← Older revision		Revision as of 16:11, 12 July 2010
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	This seems to imply that we have discovered a conserved quantity, namely, <math>(h_2^2 \dot{\lambda}_2)^2</math>. On the other hand, I might just be using a circular argument; I might only be saying that "the equation of motion is the equation of motion!"		This seems to imply that we have discovered a conserved quantity, namely, <math>(h_2^2 \dot{\lambda}_2)^2</math>. On the other hand, I might just be using a circular argument; I might only be saying that "the equation of motion is the equation of motion!"
	</font></b>		</font></b>

			Temp note: Joel, I don't quite understand this. Next time we get together, can you explain this page to me?

			==Conserved Quantity==

			Let's cut to the chase. As shown on the page describing the characteristic vector approach, I can write down the third conserved quantity right now--just not in closed form. Assuming there's no potential variation in the direction of <math>\lambda_2</math>, it is

			<div align="center">
			<math>
			m{h_2}^2 \dot{\lambda_2} \exp \left\{ - \int \frac{{h_k}^2}{{h_2}^2} \Gamma^k_{2j} \frac{\dot{\lambda_j} \dot{\lambda_k}}{\dot{\lambda_2}} \ dt \right\} .
			</math>
			</div>

			In the case of T3 coordinates, this becomes more specific.

			<div align="center">
			<math>
			m {h_2}^2 \dot{\lambda_2} \exp \left\{ - \int 2 {\lambda_1}^2 \ell^4 \left( \frac{\lambda_2 \dot{\lambda_1}^2}{\dot{\lambda_2}} - \frac{\dot{\lambda_2}{\lambda_1}^2}{\lambda_2} \right) dt \right\}
			</math>
			</div>

			Although this is not all that useful in an analytic sense until we can integrate it, I wonder if it can be a guide to building a more accurate numerical model. Certainly this function can be integrated numerically, and that's got to be useful somehow...

	<br />		<br />
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Tohline: Begin "implications" subsection

2010-06-07T01:42:51Z

Begin "implications" subsection

← Older revision		Revision as of 01:42, 7 June 2010
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	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \frac{\ln(\lambda_1 \lambda_2)}{dt} \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>
	</td>		</td>
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	So we see that, indeed, the two formalisms produce identical forms of the equation of motion.		So we see that, indeed, the two formalisms produce identical forms of the equation of motion.


			==Implications==
			Backing up to the expression that began our examination of the Binney and Tremaine formalism, we also can write,

			<table border="0" align="center" cellpadding="5">
			<tr>
			<td align="right" colspan="2">
			<math>
			\frac{d\ln(h_2^2 \dot{\lambda}_2)}{dt}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="3">
			<math>
			\frac{\lambda_2}{\dot{\lambda}_2} \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="left">
			<math>\Rightarrow</math>
			</td>
			<td align="right">
			<math>
			\frac{d\ln(h_2^2 \dot{\lambda}_2)}{dt} \biggl(\frac{d\ln\lambda_2}{dt}\biggr)
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="3">
			<math>
			\biggl[ \frac{d\ln(\lambda_1 \lambda_2)}{dt} \biggr]\frac{d\ln h_2}{dt}
			</math>
			</td>
			</tr>
			</table>

			Comparing this with the ''brute force'' derivation of the condition derived above for the characteristic vector, <math>C_2</math>, we see that the two expressions are the same if we set,
			<div align="center">
			<math>
			C_2 = h_2^2 \dot{\lambda}_2 .
			</math>
			</div>

			<b><font color="darkgreen">
			This seems to imply that we have discovered a conserved quantity, namely, <math>(h_2^2 \dot{\lambda}_2)^2</math>. On the other hand, I might just be using a circular argument; I might only be saying that "the equation of motion is the equation of motion!"
			</font></b>

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Tohline: /* Two Views of Equation of Motion */ Summary of comparison subsection

2010-06-07T01:11:07Z

Two Views of Equation of Motion: Summary of comparison subsection

← Older revision		Revision as of 01:11, 7 June 2010
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	</table>		</table>

			===Summary===
			So we see that, indeed, the two formalisms produce identical forms of the equation of motion.


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Tohline: /* Two Views of Equation of Motion */ Binney and Tremaine form of EOM

2010-06-07T01:04:21Z

Two Views of Equation of Motion: Binney and Tremaine form of EOM

← Older revision		Revision as of 01:04, 7 June 2010
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	The second component of the equation of motion can be obtained by setting <math>i = 2</math> and <math>C_i = 1</math> in '''[[User:Jaycall/KillingVectorApproach#CV.01\|Equation CV.01]]''', specifically,		The second component of the equation of motion can be obtained by setting <math>i = 2</math> and <math>C_i = 1</math> in '''[[User:Jaycall/KillingVectorApproach#CV.01\|Equation CV.01]]''', specifically,

	<table border="1" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
	<tr>		<tr>
	<td align="right">		<td align="right">
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	</tr>		</tr>
	</table>		</table>


			===Binney and Tremaine Formalism===

			We have also derived the second component of the equation of motion following the formalism outlined by Binney and Tremaine ([[User:Tohline/Appendix/References#BT87\|BT87]]). Specifically, in our introductory discussion of the [[User:Tohline/Appendix/Ramblings/T3Integrals\|T3 Coordinate System]] our '''[[User:Tohline/Appendix/Ramblings/T3Integrals#EOM.01\|Equation EOM.01]]''' has the form,

			<table border="0" align="center" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			\frac{d(h_2 \dot{\lambda}_2)}{dt}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\biggl(\frac{\lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr) \frac{dh_2}{dt} .
			</math>
			</td>
			</tr>

			</table>

			To compare this with the form derived using the Christoffel symbol formalism, we need to multiply through by <math>h_2</math> and bring the scale factor inside the time-derivative on the left-hand-side.

			<table border="0" align="center" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			\frac{d(h_2^2 \dot{\lambda}_2)}{dt}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="3">
			<math>
			\biggl[ \biggl(\frac{h_2 \lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr) + (h_2 \dot{\lambda}_2) \biggr]\frac{dh_2}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="1">
			<math>
			\biggl[ \dot{\lambda}_2 + \frac{\lambda_2 }{\lambda_1} \dot{\lambda}_1 \biggr]\biggl[ \dot{\lambda}_2 - \frac{\lambda_2}{\lambda_1} \dot{\lambda}_1 \biggr] h_2 \frac{\partial h_2}{\partial\lambda_2}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="1">
			<math>
			\biggl[ \dot{\lambda}_2^2 - \biggl( \frac{\lambda_2 }{\lambda_1}\biggr)^2 \dot{\lambda}_1^2 \biggr] h_2 \frac{\partial h_2}{\partial\lambda_2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="1">
			<math>
			\biggl( h_2 \frac{\partial h_2}{\partial\lambda_2}\biggr) \dot{\lambda}_2^2 - \biggl[\frac{h_2 \lambda_2^2}{\lambda_1^2} \dot{\lambda}_1^2 \biggr] \biggl[- \frac{h_1 \lambda_1^2}{h_2 \lambda_2^2} \frac{\partial h_1}{\partial\lambda_2} \biggr]
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="1">
			<math>
			\biggl( h_2 \frac{\partial h_2}{\partial\lambda_2}\biggr) \dot{\lambda}_2^2 + \biggl( h_1 \frac{\partial h_1}{\partial\lambda_2}\biggr) \dot{\lambda}_1^2
			</math>
			</td>
			</tr>
			</table>



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Tohline: Two views of Equation of Motion

2010-06-06T23:06:04Z

Two views of Equation of Motion

← Older revision		Revision as of 23:06, 6 June 2010
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	</tr>		</tr>

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	<td align="center">		<td align="center">
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	</td>		</td>
	</tr>		</tr>
			-->

	</table>		</table>

			==Two Views of Equation of Motion==

			===Christoffel Symbol Formalism===

			The second component of the equation of motion can be obtained by setting <math>i = 2</math> and <math>C_i = 1</math> in '''[[User:Jaycall/KillingVectorApproach#CV.01\|Equation CV.01]]''', specifically,

			<table border="1" align="center" cellpadding="5">
			<tr>
			<td align="right">
			<math>
			\frac{d(h_2^2 \dot{\lambda}_2)}{dt}
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			{h_k}^2 \Gamma^k_{2j} \dot{\lambda}_j \dot{\lambda}_k
			</math>
			</td>
			<td align="left">
			<math>
			= {h_1}^2 \dot{\lambda}_1 \biggr[ \Gamma^1_{21} \dot{\lambda}_1 + \Gamma^1_{22} \dot{\lambda}_2 \biggl] +
			{h_2}^2 \dot{\lambda}_2 \biggr[ \Gamma^2_{21} \dot{\lambda}_1 + \Gamma^2_{22} \dot{\lambda}_2 \biggl]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left" colspan="2">
			<math>
			{h_1}^2 \dot{\lambda}_1 \biggr[ \biggl( \frac{1}{h_1} \frac{\partial h_1}{\partial\lambda_2} \biggr) \dot{\lambda}_1 - \biggl( \frac{h_2}{h_1^2} \frac{\partial h_2}{\partial\lambda_1} \biggr) \dot{\lambda}_2 \biggl] +
			{h_2}^2 \dot{\lambda}_2 \biggr[ \biggl( \frac{1}{h_2} \frac{\partial h_2}{\partial\lambda_1} \biggr) \dot{\lambda}_1 + \biggl( \frac{1}{h_2} \frac{\partial h_2}{\partial\lambda_2} \biggr) \dot{\lambda}_2 \biggl]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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


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Tohline: Begin investigation of the Characteristic Vector in the specific context of T3 coordinates

2010-06-06T21:35:12Z

Begin investigation of the Characteristic Vector in the specific context of T3 coordinates

New page

{{LSU_HBook_header}}

=Characteristic Vector for T3 Coordinates=
Let's apply Jay's [[User:Jaycall/KillingVectorApproach|Characteristic Vector approach]] to Joel's [[User:Tohline/Appendix/Ramblings/T3Integrals|T3 Coordinate System]].

==Brute Force Manipulations==

Starting from '''[[User:Jaycall/KillingVectorApproach#CV.02|Equation CV.02]]''', and plugging in expressions for various [[User:Tohline/Appendix/Ramblings/T3Integrals#Logarithmic_Derivatives_of_Scale_Factors|logarithmic derivatives of the T3 scale factors]], we obtain,

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
 
</td>
<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>

<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>
</td>
</tr>

</table>

 <br />
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