Editing Jaycall/KillingVectorApproach (section)

==Killing Vector Approach==
Thanks for writing out all the terms in the expression for <math>d\Xi/dt</math> in your "[[User:Jaycall/KillingVectorApproach|Killing Vector Approach]]" discussion.  On the one hand, I'm happy that the term I mentioned cancels with another one because that means we have fewer terms to integrate.  On the other hand, it was one term that I thought we might actually be able to manipulate analytically.  Of the two terms that remain, one is relatively simple &#8212; the one that is proportional to <math>\dot{\lambda}_2</math> &#8212; but the first term is a bear!  That will take some thinking! --[[User:Tohline|Tohline]] 10:50, 30 May 2010 (MDT)

:I know it seems like the wrong direction, but unless we can express this summary integral entirely in terms of <math>\lambda_1</math>, <math>\lambda_2</math>, <math>\dot{\lambda}_1</math> and <math>\dot{\lambda}_2</math>, I think that in order to make progress we're going to have to write things out in terms of <math>\varpi</math>, <math>z</math>, <math>\dot{\varpi}</math> and <math>\dot{z}</math>.  Of course, that will make the expression much messier, but I think it will be very difficult to recognize this quantity as an exact derivative unless everything's in terms of the same coordinates. --[[User:Jaycall|Jaycall]] 12:55, 30 May 2010 (MDT)

::I agree.  It is for similar reasons that I am planning on focusing on the case where <math>q^2=2</math>.  At least in this special case we can invert the coordinate expressions to give <math>\varpi</math> and <math>z</math> in terms of the <math>\lambda</math>s.  Incidentally, I have added a link to your "Killing Vector" page as well as a link to this associated "Talk" page on the [[User:Tohline/Appendix/Ramblings|page where I itemize my various "ramblings"]]. --[[User:Tohline|Tohline]] 16:16, 30 May 2010 (MDT)

:::Good point.  That would be a good reason to focus on one specific case.  I haven't looked closely at the inverted coordinate transformation for the <math>q^2=2</math> case for several days.  Since the relation is quadratic, is it obvious which root should be used?

::The proper physical root was obvious to me when I performed the coordinate inversion in the case of [[User:Tohline/Appendix/Ramblings/T1Coordinates#Coordinate_Inversion|T1 Coordinates]], so I presume it will be obvious for the T3 Coordinate system.  But the inversion needs to be redone for the case of T3 Coordinates; do you want to take care of this and type up the result?  In the (quadratic) case of <math>q^2=2</math>, it may be as simple as replacing <math>\chi_2</math> with <math>1/\lambda_2</math>. --[[User:Tohline|Tohline]] 11:21, 31 May 2010 (MDT)

:::Sure. --[[User:Jaycall|Jaycall]] 13:34, 31 May 2010 (MDT)

'''<font color="red">Mistake?</font>''' Please check the next to last row in the [[User:Jaycall/KillingVectorApproach#Christoffel_Symbols|tabulated expressions for Christoffel symbols]]; shouldn't the <math>i</math> and <math>j</math> indexes be swapped? --[[User:Tohline|Tohline]] 09:39, 6 June 2010 (MDT)

:You are correct.  Good eye.--[[User:Jaycall|Jaycall]] 09:42, 9 June 2010 (MDT)

::Actually, it wasn't my good eye.  I discovered this from my inelegant brute-force derivations.  In particular, when I was trying to derive the same form of the [[User:Tohline/Appendix/Ramblings/T3CharacteristicVector#Two_Views_of_Equation_of_Motion|equation of motion from two different points of view]], I had to plug in some of your Christoffel symbol expressions.  At first I could not get my EOM to match yours; it was while trying to understand this mismatch that I realized that your ''general'' expression for <math>\Gamma^i_{ij}</math> did not match the ''specific'' expression for <math>\Gamma^2_{21}</math> that you typed on the "T3 Coordinates" page.  In my effort to get the EOMs to match, I decided that your ''general'' expression was the incorrect one. --[[User:Tohline|Tohline]] 14:58, 9 June 2010 (MDT)

:::For the record, on 10 June 2010, I modified the [[User:Jaycall/KillingVectorApproach#Christoffel_Symbols|table that contains the general Christoffel symbol expressions]] in order to correct the order of these indexes.--[[User:Tohline|Tohline]] 18:55, 11 June 2010 (MDT)