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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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  • $\begingroup$ Are you talking classical or quantum mechanics? In any case symmetry have generators. If a system is invariant under a certain symmetry the generator is conserved. Think of translation, which is generated by momentum. $\endgroup$
    – lcv
    May 16, 2020 at 9:07
  • $\begingroup$ What about discrete symmetries though? $\endgroup$ May 16, 2020 at 21:22
  • $\begingroup$ Good observation. An important point indeed is the fact that symmetries form a group. $\endgroup$
    – lcv
    May 16, 2020 at 21:41
  • $\begingroup$ Discrete symmetries don't correspond to conservation laws. They correspond to selection rules. $\endgroup$
    – user87745
    May 17, 2020 at 5:16
  • $\begingroup$ No you can have, for example, lattice translational symmetry which has crystal momentum conserved or even more discrete: space reversal which preserves parity $\endgroup$ May 17, 2020 at 7:20

2 Answers 2

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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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  • $\begingroup$ But how would this explain discrete symmetries? $\endgroup$ May 17, 2020 at 7:21
  • $\begingroup$ @AakashLakshmanan Quantum mechanically, you can easily imagine a generator of discrete symmetry, such as parity, to be commuting with Hamiltonian if parity is a symmetry of the Hamiltonian and then it would ensure that parity is conserved. But I don't find a good way to visualize this in terms of Hamiltonian vector fields in classical mechanics. Maybe someone who knows better can point out, it should be something very analogous. $\endgroup$
    – user87745
    May 17, 2020 at 7:33
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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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  • $\begingroup$ 1. This assumed Noether's theorems rather than explaining the reasoning/intuition behind them. 2. I'd argue this is the reverse of what's actually done. You mostly estimate the symmetries using your intuition rather than guessing conserved quantities. It can go other way around, but mostly not. $\endgroup$
    – user87745
    May 17, 2020 at 13:10
  • $\begingroup$ @Dvij D.C. I don't need Noether's theorem to know that e.g. in spherical symmetry angular momentum is conserved. $\endgroup$
    – my2cts
    May 17, 2020 at 13:49
  • $\begingroup$ You kind of do. In QM, we use the Hamiltonian formalism which makes the notion and implications of symmetries trivial. But, it is, in fact, Noether's theorem when we say that due to spherical symmetry, its generator should be conserved and should commute with Hamiltonian. $\endgroup$
    – user87745
    May 17, 2020 at 13:59
  • $\begingroup$ @Dvij D.C. The question is about physical intuition. Indeed Noether's theorem allows me to mathematically prove a conservation law, given a Lagrangian. That is not intuition. Intuition is what made me chose a Lagrangian with a specific symmetry in the first place. $\endgroup$
    – my2cts
    May 17, 2020 at 14:05
  • $\begingroup$ OP's post is specifically about how conservation laws and symmetries are related--not how we decide what symmetries to put into our Lagrangian. $\endgroup$
    – user87745
    May 17, 2020 at 14:07

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