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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.