User:Tohline/Appendix/Ramblings/T3CharacteristicVector
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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,
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<math> = </math> |
<math> \biggl[ \frac{\dot{\lambda}_1}{\lambda_1} + \frac{\dot{\lambda}_2}{\lambda_2} \biggr] \frac{d\ln h_2}{dt} </math> |
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<math> = </math> |
<math> \biggl[ \frac{d\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,
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<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> |
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<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> |
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<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> |
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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,
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<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.
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<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> |
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<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> |
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<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> |
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<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.
Implications
Backing up to the expression that began our examination of the Binney and Tremaine formalism, we also can write,
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<math> \frac{d\ln(h_2^2 \dot{\lambda}_2)}{dt} </math> |
<math>=</math> |
<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> |
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<math>\Rightarrow</math> |
<math> \frac{d\ln(h_2^2 \dot{\lambda}_2)}{dt} \biggl(\frac{d\ln\lambda_2}{dt}\biggr) </math> |
<math>=</math> |
<math> \biggl[ \frac{d\ln(\lambda_1 \lambda_2)}{dt} \biggr]\frac{d\ln h_2}{dt} </math> |
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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,
<math> C_2 = h_2^2 \dot{\lambda}_2 . </math>
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!"
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© 2014 - 2021 by Joel E. Tohline |