# General question about E-L in revativity

by noamriemer
Tags: revativity
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 P: 50 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!
 Sci Advisor Thanks P: 4,160 noamriemer, There are several different equivalent Lagrangians that may be used to calculate the geodesics. The easiest, I believe, is to use L = gμν dxμ/ds dxν/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.

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