Christoffel symbols etc. via Lagrangian

[jixe]

I believe there is a way of calculating Christoffel symbols which is easier and less time-consuming than using the metric formula directly. This involves writing down the Lagrangian in a form that just includes the kinetic energy assuming zero potential energy and then equating the coefficient of a pair of coordinates to the equivalent C symbol with those 2 coordinates as its lower indices.

I can get the Lagrangian for simple metrics, but once it starts getting a bit more complex, with multiple curvilinear coordinates etc. I'm stuffed.

Is there a foolproof procedure for writing this Lagrangian when you have to include terms theta phi, r phi, etc.? Also , when you write the C symbol with the lower values, what is the significance of the upper C symbol coordinate ( what should it be? )

Related question

How do you get geodesic equations from a metric/line element ? Once again I can see in the case of a photon you can set the line element to zero, but what about massive particles ?

I don't feel I can move on until I get this stuff sorted out and I have a sneaky feeling that it maybe just involves looking at the whole thing in a slightly different way, but while I get hung up on the detail there's no chance.

 

[robphy]

possibly helpful:
http://www.aapt-doorway.org/TGRU/articles/Moore%20GRArticle.pdf (page 8)
http://count.ucsc.edu/~rmont/classes/121A/polarChristoffelE_Lag.pdf

 

[jixe]

Thank you robphy.

I can't say I understood much of the first article, but the second was right on the button. (Though I guess he must have left out a bracket in the equation starting L=, which had me going for a while).

What I needed was a nice simple example like that.

Now I will go away and try it out a few times just to be sure. Thanks again.