1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted