Nature of Energy

[Alex K]

@little_monkey

You misunderstood. I didn't mean that energy is the most important concept in physics in general. Read again what I wrote, the sentence is: "Other conserved quantities,...but.....", so I was saying that among the conserved quantities encountered in systems, energy tends to be the most powerful one because it is encountered in most systems, and because it is often closely related to the hamiltonian which describes  the dynamics of the system.

Me, misunderstanding?? No way, 

Your Hamiltonian is important because we need it to do perturbation theory. It doesn't play much of a role in non-perturbative theories ( for instance, String Theory). The Lagrangian is far more important - it's with the Lagrangian you get your symmetries checked out, and most importantly, your Lorentz invariance is absolutely crucial  to go from QM to QFT, and you get your theory Lorentz invariant through the Lagrangian. Moreover, the people in the 50's and 60' couldn't figure out the nuclear forces, both the weak and the strong. You don't know the force you're pretty much handicapped in developing any dynamical theory. So the whole plan was: try guessing the Lagrangian - you know if you have it right, you also know you have the right equation of motion. It was a nice way to circumvent not knowing the nuclear forces, and with Yukawa's idea, we could ignore "force" and replace it with "interaction".

Working with the Lagrangian in QFT is so much more fun for the reasons you mention, no argument about that. Especially if you're willing to employ path integrals from the get-go. Still, defining the S-Matrix and showing its unitarity is a bit of a shlep based purely on Lagrangian path integrals, compared to using the Hamiltonian time evolution.

However, your comment is not relevant for my point that Energy is important conceptually because - among other things - it is often directly related to the Hamiltonian which specifies the dynamics of the system. Whether going to the Lagrangian is more advantageous for some calculations is beside the point.