General question about E-L in revativity

[noamriemer]

Hello! I ask this question in this forum because I wish to be given a general explanation.
Hope it is OK

given this metric:
$ds^2= \frac {dt^2} {t^2}- \frac{dx^2} {t^2}$
I wish to calculate the geodesics.

$S=\int{ \frac {d} {d\lambda} \sqrt{\frac {1} {t^2} \frac {dt^2} {d\lambda^2}- \frac{1} {t^2} \frac {dx^2} {d\lambda^2}}}$

But $\lambda$ here, is any parameter I choose. Therefore, L does not depend on it (right? )
So now I want to use E-L equations:

$\frac {d}{dλ}\frac {∂L}{∂\dot t}=\frac {\partial L} {∂t}, \frac {d} {dλ}\frac {∂L} {∂\dot x} =\frac {\partial L} {∂x}$

When $\dot t$ refers to $\frac {dt} {d\lambda}$ etc.
But here I get confused:
What does L depend on?
I'll continue my solution:

$\frac {d} {dλ} 0.5 ({{\frac {{\dot t}^2} {t^2}-\frac {{\dot x}^2} {t^2}}})^{-0.5} 2 \frac {\dot t} {t^2}= \frac {\partial L} {\partial x}$

As I stated, L does not depend on lambda, and so:

$\frac {d} {d\lambda} {[\frac {\dot t} {t^2}]}^2 = \frac {{\dot t}^2-{\dot x}^2} {t^3}$
So how am I supposed to derive this? Do I add another dot?
$\frac {d} {d\lambda} \dot t = \ddot t$ ? and do I refer to it as an operator? Meaning- $\ddot t = {\dot t}^2?$

How do I continue? These calculations on classical mechanics were so trivial to me- but for some reason I get lost in here...
Thank you!

 

[Bill_K]

noamriemer, There are several different equivalent Lagrangians that may be used to calculate the geodesics. The easiest, I believe, is to use L = g_μν dx^μ/ds dx^ν/ds. This eliminates dealing with the square root. So for your case, L = t^-2 (dt/ds)² - t^-2 (dx/ds)². You let t' = dt/ds and x' = dx/ds for short, and write the equations as, for example, d/ds(∂L/∂t') - ∂L/∂t = 0.