Considering a Lorentz-invariance Lagrangian for a free particle $$L=\frac{m}{2}\eta_{\nu \mu}u^{\mu}u^{\nu}$$ In the coordinates you use the Minkowski metric has constant components so the Euler-Lagrange equation: $$\frac{d}{d \tau} \left( m \eta_{\mu \nu} u^{\mu}\right)=0$$
I do not understand why. I think we need to replace $L=\frac{m}{2}\eta_{\nu \mu}u^{\mu}u^{\nu}$ in $$\dfrac{d}{dt} \left( \dfrac{ \partial L} { \partial \dot{q}^{ \lambda}} \right)- \dfrac{ \partial L}{ \partial q^{ \lambda}} = 0$$ But in the passages I got stuck. The second equality I do not understand is: $$\frac{d}{d \tau}(\eta_{\nu \mu} u^{\mu}u^{\nu})= 2 u^{\nu}\frac{d}{d \tau}(\eta_{\nu \mu} u^{\mu})=0$$
$\eta_{\mu\nu}u^\mu u^\nu$ is an invariant equal to $\pm c^2$ depending on the signature of your metric. Thus its derivative w/r to $\tau$ (which is another invariant) is automatically $0$.