I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive Newton's 2nd law of motion.
If we get Newton's second law out, does it mean that the formulation is correct? Couldn't it be just a coincidence?
Where do we derive these expressions for the Action and for the Lagrangian from?