The motivation for this problem comes from solving some complex analysis problem but it can be stated independently.
I am looking at all smooth curves joining $(0,0)$ and $(1,s)$ where $0<s<1$ and I want to find the curve which minimizes $\int_0^1 L(t,x(t),x'(t))= \int_0^1 \frac{x'(t)}{1-x(t)^2}$.
I know that the answer should be a straight line of slope $s$ (and the integral would be the hyperbolic distance between $0$ and $s$ on the real line).
The Euler-Lagrange equation (loosely stated) as $\frac{d}{dt} L_{x'} -L_{x}=0$ should have given me a differential equation whose solutions would contain my minmizers.
But that does not help since my differential equation is $0$, so no conditions. I tried to look at the variations of second order(for the straight line curve of slope $s$) but it seems that I need to prove
$ \int\limits_0^1 \frac{v}{(1+s^2x^2)^3}[(3s^2x^2-1)v-2sx(1+s^2x^2)v'] >0 $ where $v$ is the (smooth) variation curve joining $0$ and $s$ and taking value $0$ there.
I could show that integral cannot be $0$ but cannot rule out the case that it is negative.
I would be grateful if somebody could help.
Thanks