- #1

ShayanJ

Gold Member

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It turns out that if we use an affine parameter (s) to parameterize the geodesic, then the two Lagrangians become equivalent:

## \frac{d}{ds}\frac{\partial L^2}{\partial \dot x^\mu}=\frac{\partial L^2}{\partial x^\mu} \rightarrow \frac{d}{ds}\left(2L\frac{\partial L}{\partial \dot x^\mu}\right)=2L\frac{\partial L}{\partial x^\mu} \rightarrow \frac{dL}{ds}\frac{\partial L}{\partial \dot x^\mu}+L \frac {d}{ds}\frac{\partial L}{\partial \dot x^\mu}=L\frac{\partial L}{\partial x^\mu} \xrightarrow{\frac{dL}{ds}=0} \frac {d}{ds}\frac{\partial L}{\partial \dot x^\mu}=\frac{\partial L}{\partial x^\mu}##.

My problem is, ##\frac{dL}{ds}=0## means that the Lagrangian is a constant and its possible to use a parametrization in which ## L=1 ##. In this case, what does it mean to extremize the functional ## \int L^2 ds ##? Also a constant Lagrangian gives 0=0 as the EL equations of motion. What's wrong here?

Thanks