> For example, Lagrangian mechanics and Hamiltonian mechanics are equivalent to Newtonian mechanics, but they can give much better intuition for certain problems. Feynman diagrams are equivalent to grinding out the QFT algebra by hand à la Schwinger, but they give a completely different intuition for the underlying Physics.

I just read about Langrangian and Hamiltonian mechanics. I didn't encounter those at all in my EE physics, and they are fascinating. Great examples! Are you a physics professor, or is this stuff undergrad physics majors learn?