From what I’ve studied till now, Hamiltonian was initially introduced as some quantity that is usually conserved in a system and it can be total energy and it can’t either. If it is not fixed if it is total energy or not then what is the significance of even conparing it to total energy? Why not just directly use the relation of hamiltonian and lagrangian?
Edit: My question is more on the relevance of comparing Hamiltonian and total energy even though there are scenerio where both are unequal?
