User:Tohline/Appendix/Ramblings/T3CharacteristicVector

Characteristic Vector for T3 Coordinates

Let's apply Jay's Characteristic Vector approach to Joel's T3 Coordinate System.

Brute Force Manipulations

Starting from Equation CV.02, and plugging in expressions for various logarithmic derivatives of the T3 scale factors, we obtain,

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

 

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