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g

_{ab}(dx

^{a}/dτ)(dx

^{b}/dτ) = -c

^{2}

Now, I've read that you can apply this law to the Euler-Lagrange equation in order to get some equations that are apparently equivalent to the geodesic equations.

Now here is the Euler-Lagrange equation:

(∂L/∂x

^{a}) - (d/dτ)(∂L/∂(∂x

^{a}/∂τ)) = 0

Now, I have read that to find the Lagrangian L, you do this:

L = g

_{ab}(dx

^{a}/dτ)(dx

^{b}/dτ)

This however, is the exact same conservation law as mentioned in the beginning.

This would mean that L = -c

^{2}, which is just a constant.

If you plug L = -c

^{2}into the Euler-Lagrange equation, then you literally just get 0 = 0 (which is not useful at all).

How then, is the conservation law mentioned in the beginning supposed to be used with the Euler-Lagrange equation in order to solve the geodesic equations?