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Christoffel symbols etc. via Lagrangian

  1. Apr 30, 2009 #1
    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.
     
  2. jcsd
  3. Apr 30, 2009 #2

    robphy

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    Last edited by a moderator: May 4, 2017
  4. Apr 30, 2009 #3
    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.
     
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