- #1
Hamiltonian
- 296
- 190
I was following David tongs notes on GR, right after deriving the Euler Lagrange equation, he jumps into writing the Lagrangian of a free particle and then applying the EL equation to it, he mentions curved spaces by specifying the infinitesimal distance between any two points, ##x^i##and ##x^i + dx^i##, the line element as ##ds^2 = g_{ij}(x)dx^i dx^j##
and the Lagrangian for the free particle as:
$$\mathcal{L} = \frac{1}{2}mg_{ij}(x)\dot{x^j}\dot{x^i}$$
I don't understand why he has introduced the metric tensor here. He doesn't really explain how he has written the above equations and I feel a bit lost.
His handouts state clearly that you don't need previous experience with GR to follow it, am I missing something obvious?
I also don't understand how he is taking the derivatives of the the Lagrangian and then putting everything together to get the geodesic equation.
![Confused :oldconfused: :oldconfused:](https://www.physicsforums.com/styles/physicsforums/xenforo/smilies/oldschool/confused.gif)
and the Lagrangian for the free particle as:
$$\mathcal{L} = \frac{1}{2}mg_{ij}(x)\dot{x^j}\dot{x^i}$$
I don't understand why he has introduced the metric tensor here. He doesn't really explain how he has written the above equations and I feel a bit lost.
His handouts state clearly that you don't need previous experience with GR to follow it, am I missing something obvious?
I also don't understand how he is taking the derivatives of the the Lagrangian and then putting everything together to get the geodesic equation.
![Confused :oldconfused: :oldconfused:](https://www.physicsforums.com/styles/physicsforums/xenforo/smilies/oldschool/confused.gif)
Last edited: