Alternate form of geodesic equation

[Alexrey]

Homework Statement

We're asked to show that the geodesic equation $$\frac{du^{a}}{dt} +\Gamma^{a}_{bc}u^{b}u^{c}=0$$ can be written in the form $$\frac{du_{a}}{dt}=\frac{1}{2}(\partial_{a}g_{cd})u^{c}u^{d}$$

Homework Equations

$$\Gamma^{a}_{bc}=\frac{1}{2}g^{ad}(\partial_{b}g_{dc}+\partial_{c}g_{bd}-\partial_{d}g_{bc})$$

The Attempt at a Solution

I tried contracting the geodesic equation with $$g_{ab}$$ but came out with some Kronecker deltas which stumped me a bit.

 

[Oxvillian]

It can't be true in general. (pun!)

Is there some kind of simplifying assumption or approximation that makes 2 of the terms go away? Perhaps you're studying the Newtonian limit?