# Variational Calculus

1. Feb 25, 2012

### MarkovMarakov

1. The problem statement, all variables and given/known data
The Riemann metric on $\{z\in C: |z|<1\}$ is defined as $dx^2+dy^2\over 1-(x^2+y^2)$. 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