Coordinate free derivation of Euler-Lagrange equations

Question

I am studying Symplectic Geometry and I was wondering how one can compute Euler-Lagrange equation in a coordinate free manner. For instance, I know for the following Lagrangian $L(x,v)=\frac{1}{2}g_{x}(v,v)-V(x)$ the corresponding equation is $\nabla_{\dot\gamma}\dot\gamma=-\nabla V$, but it is a bit complicated to derive it in a coordinate system.