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## Main Question or Discussion Point

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.

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?

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.Particularly it enables us to formulate symmetry principles in a mathematically elegant way (Lie-group theory), and in this way it reveals a one-to-one relationship between symmetries and conservation laws (Emmy Noether 1918). In this way it also enables us to calculate the fundamental quantities in any model based on the symmetries of Newtonian, Minkowski or psesuo-Riemannian spacetimes as the generators of the fundamental symmetries of the spacetime geometry, i.e., if there is translation invariance, you can define momentum as the generator of the corresponding translation transformation on the space-time variables, etc.

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?