In the case of classical mechanics, when describing the movement of a particle that is subject to a force, Newton's law gives us the law of motion of the particle: $$F(t)=m\cdot a(t)$$
We can also apply the Lagrangian instead: $$L=\frac 1 2 m\cdot v^2 - V(x)$$ Applying the principle of least action by the Euler-Lagrange equation we get: $$\frac d {dt}\frac {dL}{dv}=\frac {dL}{dx}$$ Which simply reduces to Newton's equation.
My question is: If minimizing the action simply reduces to finding Newton's equation, then what is the added value of the lagrangian in the first place? Why not just start with Newton's equations?
