- #1
Nabigh R
- 11
- 0
I am trying to derive the geodesic equation by extremising the integral
$$ \ell = \int d\tau $$
Now after applying Euler-Lagrange equation, I finally get the following:
$$ \ddot{x}^\tau + \Gamma^\tau_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = \frac{1}{2} \dot{x}^\tau \frac{d}{ds} \ln \left| \dot{x}_\nu \dot{x}^\nu \right| $$
where ## \dot{x}^\tau \equiv \frac{d x^\tau}{ds} ## and ##s## is a parameter. Now I get the geodesic equation if the right-hand side vanishes, and the only way that happens is if ## \dot{x}_\nu \dot{x}^\nu ## is constant. Now the question is why is it constant?
$$ \ell = \int d\tau $$
Now after applying Euler-Lagrange equation, I finally get the following:
$$ \ddot{x}^\tau + \Gamma^\tau_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = \frac{1}{2} \dot{x}^\tau \frac{d}{ds} \ln \left| \dot{x}_\nu \dot{x}^\nu \right| $$
where ## \dot{x}^\tau \equiv \frac{d x^\tau}{ds} ## and ##s## is a parameter. Now I get the geodesic equation if the right-hand side vanishes, and the only way that happens is if ## \dot{x}_\nu \dot{x}^\nu ## is constant. Now the question is why is it constant?