It is clear from a Lagrangian formalism, that various types of symmetries of a system give rise to many interesting conserved properties of the given system but is there an interesting physical intuition behind? Is there some sort of physical discussion that can be had about why symmetries would intuitively conserve anything at all?
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I'm not sure if there's a direct physical intuition behind it in Lagrangian formalism but in Hamiltonian formalism, Noether theorem is almost trivial.
If the Poisson bracket between two observables vanishes then each of the observables remain invariant along the integral curve of the other observable. Now, consider one of the observables to be the Hamiltonian. If the Hamiltonian enjoys a symmetry, its Poisson bracket with the generator of the symmetry would vanish and this, in turn, would ensure that the generator of symmetry is constant along the integral curve of the Hamiltonian (i.e., it would be a constant of motion).
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First you use your physical intuition to estimate which quantities are conserved and which symmetries are involved and then you write down a suitable Lagrangian which has these symmetries.
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