Editing Jaycall/KillingVectorApproach (section)

==Two Views of Equation of Motion==

Jay: I have created a [[User:Tohline/Appendix/Ramblings/T3CharacteristicVector|new wiki page]] to explicitly analyze how your Characteristic Vector formalism applies to the T3 Coordinate system.  On this page is a subsection entitled "[[User:Tohline/Appendix/Ramblings/T3CharacteristicVector#Two_Views_of_Equation_of_Motion|Two Views of Equation of Motion]]" in which I have shown that the <math>2^\mathrm{nd}</math> component of the equation of motion can be written in the following form,
<div align="center">
<math>\frac{d(h_2^2 \dot{\lambda}_2)}{dt} = \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> ,
</div>
assuming that <math>\partial\Phi/\partial\lambda_2 = 0</math>.  More to the point, I have shown that I can derive this form of this component of the equation of motion using either (A) your Christoffel symbol formalism, or (B) the more classical formalism that I have been using, which I obtained from Appendix 1.B of Binney &amp; Tremaine (1987) and Morse &amp; Feshbach (1953).  

Now, I shouldn't be shocked that both derivations lead to the same form of the equation &#8212; after all, we're doing physics aren't we?  But here is what surprised me:  The Christoffel symbol formalism produces this form of the equation without ever assuming ''a particular coordinate system'' whereas, in order to derive the equation using the more classical formalism, I plugged in some relations between the partial derivatives of the scale factors that ''I thought'' were specific to T3 coordinates.  This leads me to ask, "How many of the T3 Coordinate relations [[User:Tohline/Appendix/Ramblings/T3Integrals|derived earlier]] are generic relations that apply to any orthogonal, axisymmetric coordinate system, and how many are unique to the T3 Coordinate System?"  I'm particularly interested in knowing how generalizable the relations are that appear inside boxes labeled "T3 Coordinates" under the subsections I have entitled, "[[User:Tohline/Appendix/Ramblings/T3Integrals#Derived_Identity_for_T3_Coordinates|Derived Identity for T3 Corodinates]]" and "[[User:Tohline/Appendix/Ramblings/T3Integrals#Logarithmic_Derivatives_of_Scale_Factors|Logarithmic Derivatives of Scale Factors]]". Can you answer this? --[[User:Tohline|Tohline]] 10:46, 7 June 2010 (MDT)

:Joel:  Yes, this is the power of using a covariant formulation.  The only assumptions we made were that the coordinates were orthogonal and axisymmetric.  I believe that the relation under "[[User:Tohline/Appendix/Ramblings/T3Integrals#Derived_Identity_for_T3_Coordinates|Derived Identity for T3 Coordinates]]" would apply to any coordinate system meeting these criteria.  I will double check, though.  (How did you determine that the position vector could be written <math>\vec{x} = \hat{e}_1 (h_1 \lambda_1) + \hat{e}_2 (h_2 \lambda_2)</math>?)

:The logarithmic derivatives of the scale factors cannot be general in their current form due to the appearance of factors of <math>q</math>, but I suspect they can be generalized and molded into a form that closely resembles what you have.  I will work on this, too. --[[User:Jaycall|Jaycall]] 10:39, 9 June 2010 (MDT)

::Jay: You asked how I determined the expression for the [[User:Tohline/Appendix/Ramblings/T3Integrals#Definition|position vector]].  I don't have my detailed notes with me right now, but I can outline the steps.  In Morse &amp; Feshbach's classical presentation of orthogonal curvilinear coordinates, they define a set of so-called ''direction cosines,'' <math>\gamma_{ij}</math>.  (I don't have the generalized definition with me at this time, but they are expressible in terms of derivatives of the scale factors with respect to the various coordinates; I'm certain ''direction cosines'' are definable in terms of Christoffel symbols.) Each of the Cartesian unit vectors (<math>\hat{\imath},\hat{\jmath},\hat{k}</math>) can be written in terms of products of these direction cosines and the three unit vectors of your chosen curvilinear coordinate system (<math>\hat{e}_1, \hat{e}_2, \hat{e}_3</math>).  According to Morse &amp; Feshbach, for example, <math>\hat{\imath} = (\hat{e}_1\gamma_{11} + \hat{e}_2\gamma_{21} + \hat{e}_3\gamma_{31})</math>.  So, all I did to obtain an expression for the position vector in T3 coordinates was define <math>\vec{x} = (\hat\imath x + \hat\jmath y + \hat{k}z)</math>, then replace each Cartesian unit vector by its expression in terms of products of the T3-coordinate unit vectors &amp; the appropriate direct cosine, then gather together and simplify terms.  Does this make sense? --[[User:Tohline|Tohline]] 14:59, 10 June 2010 (MDT)

:::Yes, it does.  That's very helpful. --[[User:Jaycall|Jaycall]] 09:49, 11 June 2010 (MDT)

::::Jay:  For the record, I have put together a wiki page that discusses the definition and utility of [[User:Tohline/Appendix/Ramblings/DirectionCosines|Direction Cosines]].--[[User:Tohline|Tohline]] 18:47, 3 July 2010 (MDT)

Joel, I finally figured out what was going on this afternoon with the strange conserved quantities we were getting, like <math>{h_2}^4 \dot{\lambda_2}^2</math> and, consequently, <math>\lambda_1 \lambda_2</math>.  There ''is'' a mistake in the derivation stemming all the way back from Eq. CV.02 on my page on the [[User:Jaycall/KillingVectorApproach|characteristic vector approach]].  My derivation of the brute force condition on <math>C_2</math> was correct, but you can see from above that <math>\Xi</math> does not equal <math>\ln C_2</math>, it equals <math>- \ln C_2</math>.  Consequently, the conclusion is no longer that <math>C_2</math> must equal <math>{h_2}^2 \dot{\lambda_2}</math>, but rather <math>\left( {h_2}^2 \dot{\lambda_2} \right)^{-1}</math>.  So in fact you ''did'' find a conserved quantity--<math>1</math>!   : )  I guess it was a tautology after all; your intuition was right.  --[[User:Jaycall|Jaycall]] 18:50, 14 July 2010 (MDT)

:Jay, very good catch!  Looking back through the history of that wiki page, it is clear that I am the one who introduced this sign error.  I mistakingly set <math>\Xi = \ln C_2</math> in equation [[User:Jaycall/KillingVectorApproach#CV.02|CV.02]] at the same time that I "Added a couple of right-justified equation numbers."  I'll march through the relevent wiki pages today and attempt to correct this mistake wherever it appears. --[[User:Tohline|Tohline]] 08:30, 15 July 2010 (MDT)

::Joel, I'm sorry yesterday's excitement didn't pan out.  It would have been really neat.  Well, at least I learned something.  On my drive home yesterday I was thinking about what it meant for there to be a constraint that involved only the coordinates (and not their rates of change).  We were uncomfortable with this idea, but now I understand why in general no conserved quantity can take this form.  If there were a constraint on only the coordinates, then this would limit a particle to a curve or a surface...a subspace of physical space (as opposed to a subspace of phase space).  Yet it would definitely be possible to get a particle to leave the subspace by giving it an initial velocity in any direction normal to the subspace.  Any conserved quantity must definitely involve the coordinate velocities (unless you're artificially restricting the motion). --[[User:Jaycall|Jaycall]] 14:15, 15 July 2010 (MDT)

:::Good observation. --[[User:Tohline|Tohline]] 16:28, 15 July 2010 (MDT)