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Noether’s theorem roughly states that the existence of a symmetry group for a given system implies a conservation law for that system. All well and good, except that I’m shaky on exactly how you predict what the conservation law is given the symmetry group. Common examples are translation symmetry implying conservation of momentum, rotational symmetry implying conservation of angular momentum, and time symmetry implying conservation of mass-energy. The pattern I see is that in a rough sense if you apply a change $\mathrm{d}q$ to a system that has $q$-symmetry, then the quantity $m\frac{\mathrm{d}q}{\mathrm{d}t}$ is conserved:

  • Systems with translation symmetry are invariant for any change in position $\mathrm{d}x$, and momentum $p=m\frac{\mathrm{d}x}{\mathrm{d}t}$ is conserved.
  • Systems with rotational symmetry are invariant for any change in angle $\mathrm{d}\theta$, and angular momentum $L\propto m\frac{\mathrm{d}\theta}{\mathrm{d}t}$ is conserved (yes I know angular momentum is more complex than that, but that’s the best my brain can do).
  • Systems with time symmetry are invariant for any change in time $\mathrm{d}t$, and mass $m\frac{\mathrm{d}t}{\mathrm{d}t}=m=\frac{E}{c^2}$ is conserved.

That’s probably all mostly incorrect but it’s the best I can do. What I’m asking is is there a straightforward way to determine what quantities are conserved if you know under what symmetries a system is invariant?

Qmechanic
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controlgroup
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1 Answers1

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  1. Yes, given a quasisymmetry of the action, Noether's theorem (NT) yields a formula for a current $J^{\mu}$ satisfying an on-shell continuity equation $d_{\mu}J^{\mu}\approx 0$ and a conserved charge $Q=\int\! d^3 J^0$, known as the Noether current and the Noether charge, respectively, see e.g. this Phys.SE post.

  2. In plain English, the quasisymmetry variable and the Noether charge constitute a canonical pair, codified in the Hamiltonian formulation of NT, cf. e.g. this Phys.SE post.

  3. For an instructive example of how to apply NT, see see e.g. this Phys.SE post.

Qmechanic
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