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Variational Calculus

  1. Feb 25, 2012 #1
    1. The problem statement, all variables and given/known data
    The Riemann metric on [itex]\{z\in C: |z|<1\}[/itex] is defined as [itex]dx^2+dy^2\over 1-(x^2+y^2)[/itex]. I wish to show that the geodesics are diameters. Please help!


    2. Relevant equations
    As above. And I suspect the Euler Lagrange equations.


    3. The attempt at a solution
    I have tried using the Euler Lagrange equations (since one way to do this is clearly to just variational calculus) but I can't get the correct form no matter what :( I even tried changing the metric into polar coordinates but I still don't get the answer.

    Also, I have noticed that this metric is very similar to the one for the hyperbolic Poincare disc model, only scaled. Is it possible to directly deduce results from that? Nonetheless I am guessing that the calculus method should be more robust.
    Thank you.
     
    Last edited: Feb 25, 2012
  2. jcsd
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