We can think of a particle having kinetic and potential energy, T and V. The Hamiltonian is the sum of these, H = T + V. This seems like a sensible enough quantity to think about. However, we can also define the Lagrangian as being the difference between these two quantities, L = T-V. However, I do not quite understand why this is an interesting/relevant/useful quantity, I don't understand what Lagrange's motivation was to start from this point. We can then think about another quantity called the action, defined as S = ∫L.dt. The action supposedly takes different values for the different trajectories that a moving particle could follow. Now apparently, classical mechanics postulates that the actual path the particle takes is the one for which the action is minimized. Why is this the case? Looking at that integral, that says to me that classically, L should be as small as possible. Since L = T-V, then that is equivalent to saying that in a classical situation, the particle should have its potential and kinetic energies being as similar as possible. Where does this come from? I don't get that at all, why should one expect that? What is the logic behind imposing this condition? This has been nagging me for a while because I'm blindly following Lagrangian mechanics to solve problems, but I don't understand its foundations. I know that it is a useful method, eg. you can easily derive Newton's laws starting from the Lagrangian, but I need to know what was going through Lagrange's head when he first dreamt it up.