We have recently covered the Lagrangian in our lectures, whereby it was shown that all equations of motion ($x(t)$) satisfying the Euler-Lagrange equation with Lagrangian $L=T-V$, where $T=\frac{1}{2}mv^2$ and $F=-\frac{\partial V}{\partial x}$, must also satisfy $F=m\ddot{x}$.
However, to me, it wasn't intuitive at all why this must have been the case: why, if we minimise (or maximise) some arbitrary quantity $L$, which didn't even have to be conserved, we for some reason get Newton's Second Law, which also implies that $L$ is in fact energy and has to be conserved.
Is there any intuitive way of understanding why ALL possible types of motion MUST minimise this arbitrary quantity $S$ and why, if we do minimise it, we get that $L$ is conserved and happens to represent the total energy of an object?
