Derivation of the Geodesic Equation from Hamilton-Jacobi Theory

Question

I'm trying to prove a result in my general relativity class and I'm confused. If we have the Hamiltonian: $$ H = g^{\mu\nu}p_\mu p_\nu + m^2 $$ Subject to Hamilton's Equations: $$\frac{\partial x^\mu}{\partial \tau} = N(\tau) \frac{\partial H}{\partial p_\mu}$$ and: $$\frac{\partial p_\mu}{\partial \tau} = - N(\tau) \frac{\partial H}{\partial x^\mu}$$ Where $N(\tau)$ is just some function of $\tau$. I'm trying to prove that: $$\frac{\partial^2 x^\mu}{\partial \tau^2} + \Gamma^\mu{}_{\alpha\beta} \frac{\partial x^\alpha}{\partial \tau}\frac{\partial x^\beta}{\partial \tau} = \text{(something)} \frac{\partial x^\mu}{\partial \tau}$$ Such that the equation can be reparametrized so that (something) = 0 and we have the geodesic equation. What's primarily tripping me up is writing: $$ \Gamma^\mu{}_{\alpha\beta} \frac{\partial x^\alpha}{\partial \tau}\frac{\partial x^\beta}{\partial \tau} $$ as proportional to $\frac{\partial x^\mu}{\partial \tau}$ What I have so far is $$\Gamma^\mu{}_{\alpha\beta} \frac{\partial x^\alpha}{\partial \tau}\frac{\partial x^\beta}{\partial \tau} = \frac{N(\tau) p_\nu}{2}g^{\mu\gamma}\left(g_{\gamma\alpha,\beta} + g_{\beta\gamma, \alpha} - g_{\alpha\beta,\gamma} \right) g^{\alpha\nu} \frac{\partial x^\beta}{\partial \tau}$$ So I guess my question is, is there a way to write: $$ g^{\mu\gamma}\left(g_{\gamma\alpha,\beta} + g_{\beta\gamma, \alpha} - g_{\alpha\beta,\gamma} \right) g^{\alpha\nu} \frac{\partial x^\beta}{\partial \tau}$$ Such that the indices $\beta$ and $\mu$ are switched, or such that this reduces in a nice way? Or have I been going about this problem incorrectly?