geodesic supplement

geodesic supplement - Supplement on the geodesic equation...

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Unformatted text preview: Supplement on the geodesic equation Im writing up this brief supplement because I noticed a lot of people get- ting things wrong when finding the geodesic equation via Hartles Lagrangian method. This method is a little tricky since there are a few steps you must follow that he doesnt explicitly state. This is best illustrated by example: Consider the hyperbolic plane (problem 8.12 on the last hw): dS 2 = y- 2 ( dx 2 + dy 2 ) Since geodesics are paths of extremal length, we want to extremize S = dS = 1 y dx d 2 + dy d 2 d where we have parametrized our path by some paramter - its not important what this parameter is since we will switch to an affine parameter during the calculation (rememeber an affine parameter is just one that make the RHS of the geodesic equation 0). Now, if we identify L = 1 y dx d 2 + dy d 2 = dS d then finding geodesics looks a lot like solving the Euler-Lagrange equations from classical mechanics....
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geodesic supplement - Supplement on the geodesic equation...

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