Here is a quote from Vanhees 71 in another thread on Lagrangians. I reposted here as a new thread because I fear going off-topic and redirecting a thread. In any case, in my study of Lagrangians and Hamiltonians, everywhere I go for tutelage it seems as though everyone is maniacally focused on symmetries. What happens if you move this a little, what happens if you move that a little. Rotational symmetry, translational symmetry, epsilons, deltas, etc. My problem is that I'm willing to buy the fact that it is important, but I'm not getting the intuition as to why. It seems as though this fixation on what happens when you change something a little bit just seems to be a preoccupation that each next generation of physicists feel they need to be neurotic about because their mentors were. Of course I jest a little, but the fact is that it is not getting through to me on an intuitive basis what these symmetry properties are or why they are so important. Perhaps someone can provide some intuition?