Noether’s second theorem: about the action principle

Question

Noether's second theorem is supposed to show that the invariance of the Lagrangian by the Lie group (infinite in dimension) of certain theories necessarily implies that the field equations proper to these theories satisfy identities, rather than "classical" conservation laws (called "proper conservation laws" by Hilbert), and the quantities thus "improperly" conserved are associated with local symmetries -- and no longer global symmetries, as in Noether's first theorem.

My question is based on a quotation from 2018/2019 lecture notes by Fiorenzo Bastianelli entitled "On Noether's theorems and gauge theories in Hamiltonian formulation" (PDF). Here is the extract:

This brief treatment shows that, when a local group of transformations has a non-trivial global subgroup, it is possible to find some conserved quantities without requiring Euler- Lagrange equations of motion to be satisfied and where arbitrary functions are involved. These conservation laws are called improper laws of conservation and we can see the seeds of gauge theories hiding in them.

(Emphasis added).

It's the bit in italics that worries me. Does the second theorem imply the existence, at least as a possibility, of movements that do not respect Euler-Lagrange's equations? In other words, violating the principle of stationary action? This is no doubt a naive question (I think I know that you can derive the GR field equation from the action principle), but one that I can't manage to solve on my own.

Answer 1

1. First of all, it should stressed that the notion of (quasi)symmetry refers to an off-shell$^1$ (quasi)symmetry. An on-shell (quasi)symmetry is basically a trivial notion. In this sense, the Noether's theorems do only assume that the physical system has an action formulation; they do not a priori assume that the Euler-Lagrange (EL) equations hold.

2. Both the first and second Noether's theorem give rise to an off-shell Noether identity. However, their form differ substantially:

3. For a global quasisymmetry in Noether's first theorem, the Noether identity turns into a continuity equation on-shell with a corresponding Noether charge conserved on-shell.

4. For a local/gauge quasisymmetry in Noether's second theorem, the corresponding Noether charge vanishes off-shell, cf. e.g. this related Phys.SE post.

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$^1$ The word on-shell means that the Euler-Lagrange (EL) equations are satisfied. The word off-shell means that the Euler-Lagrange (EL) equations are not necessarily satisfied.