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Geodesic Upper Half Plane help

  1. May 31, 2012 #1
    The metric is [tex]ds^2=\frac{dx^2+dy^2}{y^2}.[/tex] I have used the Euler-Lagrange equations to find the geodesics, and my equations are [tex]\dot{x}=Ay^2,[/tex] [tex]\ddot{y}+\frac{\dot{x}^2-\dot{y}^2}{y}=0.[/tex] I cannot seem to find the first integral for the second equation. I know it is [tex]\dot{y}=y\sqrt{1-Ay^2},[/tex] but I can't seem to derive it. The only trick I currently know for doing these type of things is to multiple by [tex]\dot{y}[/tex] and then integrate, but that doesn't work here. Can anyone offer some guidance?

    I tried it a slightly different way, but it doesn't seem to work for some reason:
    Instead of parametrizing, x=x(t), y=y(t) and minimizing, I just minimized [tex]\frac{1+y'(x)^2}{y^2}.[/tex] Using the Euler-Lagrange equations, I get [tex]y''y-y'^2+1=0,[/tex] and [tex]y(x)=sinh(x)[/tex] is a solution to this...but the geodesics are suppose to be half circles, and this doesn't give me a half circle..I am quite confused.
     
  2. jcsd
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