Is there a fundamental misunderstanding here or have I made an algebraic slip?
I have a Riemannian metric of the form $ds^2={du^2+dv^2\over 1-u^2-v^2}$ on an open disc and I want to prove that radial curve $(r(t), 0)$ is a geodesic.
I wrote the metric in polar form -- $ds^2={dr^2+r^2d\theta^2\over 1-r^2}$, so the coefficients of the first fundamental form are $E={1\over 1-r^2}, F=0, G={r^2\over 1-r^2}$.
So the geodesic equations become ${d\over dt} (E\dot{r})={1\over 2}{\partial \over \partial r} (E\dot{r}^2)$ and (another equation which works).
I can always reparametrise $r(t)$ so that it has unit speed.
And I get $2r\over (1-r^2)^2$= $r\over (1-r^2)^2$, which is clearly wrong! Why is there an extra factor of 2?
Thank you.
ADDED: Perhaps it would be helpful for me to point out that generally, for a Riemannian metric of the form $Edu^2+2Fdudv+Gdv^2$ and geodesic $g(t)=(a(t),b(t))$, the Euler Lagrange equations are
${d\over dt} (E\dot{a}+F\dot{b})={1\over 2} (E_u\dot{a}^2+2F_u\dot{a}\dot{b}+G_u\dot{b}^2)$ and
${d\over dt} (F\dot{a}+G\dot{b})={1\over 2} (E_v\dot{a}^2+2F_v\dot{a}\dot{b}+G_v\dot{b}^2)$
where $\dot{a}={d\over dt}a,\,\,\,E_u={\partial\over\partial u}E$ and so on.
EUREKA! Ooh, I think I know what the bug is, The geodesic has constant speed, but the inner product here is not simply the Euclidean one which is what I assumed when I set $\dot{r}=1$! I think it works fine now. :) -- I write it here because I can't post it as an answer...