But simply getting to the Lagrangian picture from the Hamiltonian picture would just leave me wondering why the Hamiltonian picture works!
My motivation for getting to the bottom of all this is to fill the gaps in my physics understanding at least up to quantum mechanics. I have a grasp of QM but I would like to have some insight into the conceptual leaps that brought us there from classical mechanics. QM works in the Hamiltonian picture and I recall from my undergrad days that you get there from a Legendre transformation on the Lagrangian (or something to that effect) so I'm trying to understand the justification of that approach before moving up the conceptual ladder.
Ideally I would like to be able to trace my way from simple postulates based on observation of the physical world all the way to QM, then maybe to QFT after that.
My motivation for getting to the bottom of all this is to fill the gaps in my physics understanding at least up to quantum mechanics. I have a grasp of QM but I would like to have some insight into the conceptual leaps that brought us there from classical mechanics. QM works in the Hamiltonian picture and I recall from my undergrad days that you get there from a Legendre transformation on the Lagrangian (or something to that effect) so I'm trying to understand the justification of that approach before moving up the conceptual ladder.
Ideally I would like to be able to trace my way from simple postulates based on observation of the physical world all the way to QM, then maybe to QFT after that.