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geodesic supplement

# geodesic supplement - Supplement on the geodesic equation...

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Supplement on the geodesic equation I’m writing up this brief supplement because I noticed a lot of people get- ting things wrong when finding the geodesic equation via Hartle’s “Lagrangian” method. This method is a little tricky since there are a few steps you must follow that he doesn’t 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 σ 2 + dy d σ 2 d σ where we have parametrized our path by some paramter σ - it’s not important what this parameter is since we will switch to an a ffi ne parameter during the calculation (rememeber an a ffi ne parameter is just one that make the RHS of the geodesic equation 0). Now, if we identify L = 1 y dx d σ 2 + dy d σ 2 = dS d σ then finding geodesics looks a lot like solving the Euler-Lagrange equations from classical mechanics.

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