general relativity - Finding geodesics: Lagrangian vs Hamiltonian

I have a question referring to how to compute geodesics of a given spacetime (say, Kerr).

I know that the direct way is via the geodesic equation

$$\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma^{\mu}_{\nu\kappa}\dot{x}^{\mu}\dot{x}^{\kappa}~=~0.$$

But also read that one can write down the geodesic equations using the Lagrangian formalism. From what I have seen so far, there are two approaches: either to write down the Euler-Lagrange equations with Lagrangian

$$L~=~\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}.$$

and then Euler-Lagrange equations are exactly the geodesic equations.

OR: write down the Hamiltonian equations using the Lagrangian above, and then use those equations as the geodesic equations.

My question: are Euler-Lagrange and Hamiltonian approaches fully equivalent when it comes to writing down the geodesic equations? Does one have advantage over the other?