Is there a difference, conceptually speaking, between solving the geodesic equations using $\lambda$ as an arbitrary parameter vs substituting a coordinate from the metric in it's place?
For instance, if I have:
$\frac{d^2\theta}{d\lambda^2}=-\Gamma^{\theta}_{t\theta}\frac{dt}{d\lambda}\frac{d\theta}{d\lambda}-\Gamma^{\theta}_{p\theta}\frac{dp}{d\lambda}\frac{d\theta}{d\lambda}$, and I decide $\lambda=\theta$, I get $0=-\Gamma^{\theta}_{t\theta}\frac{dt}{d\theta}-\Gamma^{\theta}_{p\theta}\frac{dp}{d\theta}$, which I can multiply by $d\theta$ and then integrate for $t$ and $p$. But I know the answer must not be true for all geodesics, so I have to ask, what does using $\theta$ as $\lambda$ specify?
By the way, if it would help to explicity see what the metric and Christoffel symbols are, I could include those.