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>

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 Equation CV.01, specifically,

<math> \frac{d(h_2^2 \dot{\lambda}_2)}{dt} </math>	<math>=</math>	<math> {h_k}^2 \Gamma^k_{2j} \dot{\lambda}_j \dot{\lambda}_k </math>	<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>
	<math>=</math>	<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>
	<math>=</math>	<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>

Binney and Tremaine Formalism

We have also derived the second component of the equation of motion following the formalism outlined by Binney and Tremaine (BT87). Specifically, in our introductory discussion of the T3 Coordinate System our Equation EOM.01 has the form,

<math> \frac{d(h_2 \dot{\lambda}_2)}{dt} </math>

<math>=</math>

<math> \biggl(\frac{\lambda_2 \dot{\lambda}_1}{\lambda_1}\biggr) \frac{dh_2}{dt} . </math>

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.

<math> \frac{d(h_2^2 \dot{\lambda}_2)}{dt} </math>	<math>=</math>	<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>
	<math>=</math>	<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>	<math>=</math>	<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>
	<math>=</math>	<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>	<math>=</math>	<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>
	<math>=</math>	<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>	<math>=</math>	<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>

Summary

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

 

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation