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If I have a massive particle constrained to the surface of a Riemannian manifold (the metric tensor is positive definite) with kinetic energy $$T=\dfrac 12mg_{\mu\nu} \dfrac{\text dx^{\mu}}{\text dt} \dfrac{\text dx^{\nu}}{\text dt}$$ then I believe I should be able to derive the geodesic equations for this manifold by applying the Euler-Lagrange equations to the Lagrangian $$L:=g_{\mu \nu}\dfrac{\text dx^{\mu}}{\text dt} \dfrac{\text dx^{\nu}}{\text dt}.$$ However, when I go to do this, here's what I find: $$\dfrac{\partial L}{\partial x^{\sigma}} = \dfrac{\partial g_{\mu\nu}}{\partial x^{\sigma}} \dfrac{\text dx^{\mu}}{\text dt}\dfrac{\text dx^{\nu}}{\text dt}.$$ Moreover, $$\dfrac{\text d}{\text dt}\left(\dfrac{\partial L}{\partial (\text dx^{\sigma}/\text dt)}\right)=\dfrac{\text d}{\text dt}\left(2g_{\sigma\mu} \dfrac{\text dx^{\mu}}{\text dt}\right)=2g_{\sigma\mu} \dfrac{\text d^2x^{\mu}}{\text dt^2}.$$ Setting these expressions equal and multiplying by the inverse metric, I obtain $$\dfrac{\text d^2x^{\tau}}{\text dt^2} - \dfrac 12 g^{\tau\sigma}\dfrac{\partial g_{\mu\nu}}{\partial x^{\sigma}} \dfrac{\text dx^{\mu}}{\text dt} \dfrac{\text dx^{\nu}}{\text dt} = 0.$$ This looks similar to the geodesic equation, but something is off about the "Christoffel Symbols" of this equation.

What's wrong with my derivation? Any help is appreciated. Thanks.