On-shell and off-shell

Plain-language counterpart: True everywhere against true for real motions.

A large outlined region labelled all configurations, containing a smaller shaded region labelled solutions. The Noether identity is annotated as holding over the whole large region; the conservation law only over the inner one. — On-shell and off-shell as nested regions

Two statements have been in play and they are not the same statement. This page separates them, because almost every compressed account of the theorem runs them together, and because the second theorem lives entirely on the side that gets dropped.

The two statements

Off-shell. The relations (N13) are identities. They hold at every point, for all values of the arguments, whether or not the $u$ satisfy anything. They are consequences of the invariance of the integral and of nothing else. Physics calls this holding off-shell.

On-shell. Pass to the associated variational problem — set $ψ_{i} = 0$ . The left-hand sides of (N13) vanish and the relations become

$D i v B^{(1)} = 0, \dots, D i v B^{(ϱ)} = 0 (1)$

These are conservation laws. In one independent variable they integrate to $B^{(λ)} = c o n s t$ ; in several they are the divergence equations that, Noether observes, had recently often been called Erhaltungssätze.

The trade

The identity is stronger in scope and weaker in content. It is true of every configuration, and it says only that a combination of Lagrange expressions is a divergence. The conservation law is true only of solutions, and it says something a physicist can measure.

Neither statement contains the other. Deriving the second from the first costs the field equations; the first cannot be recovered from the second, because the second says nothing about configurations that are not solutions.

Noether performs this passage explicitly and separately, twice — once in her first section when she surveys what the theorems mean, and once at the end of her third section. She does not fold it into the theorem. Treatments that state the theorem on-shell from the start have deleted a step she took care to keep visible.

Why the distinction is load-bearing

Three consequences, and the third is the reason this page exists.

The converse needs the identity. Running the theorem backwards starts from (N13) and reconstructs the invariance. It starts from the identity, not from the conservation law, and there is no route back from the on-shell statement alone.

The sourced case needs the identity. Setting $ψ_{i} = T_{i}$ rather than zero leaves a relation with a source term, as on the previous page. That is only available because the underlying statement was never conditioned on $ψ = 0$ .

The second theorem never crosses over. Its output is a set of identities among the Lagrange expressions and their derivatives, holding before any field equation is imposed. Setting $ψ_{i} = 0$ does not turn them into conservation laws; it turns them into the statement that some of the field equations follow from the others. A reader who has learned the first theorem only in its on-shell form has no slot to put the second theorem in, and this is the main reason the second theorem is so often mis-stated as a variant of the first.

A note on the phrase "on-shell"

The term is later than Noether and comes from the mass shell of relativistic particle kinematics. She has no term for it; she says what she is doing, which is passing to the variational problem, or setting $ψ = 0$ . The modern term is used in this volume because it is standard, and is recorded here as an anachronism rather than as her vocabulary.

Provenance: the passage to the variational problem and the note that the resulting divergence equations were then often called Erhaltungssätze are Noether's, from her first and third sections; display (1) is hers, set unnumbered in the original. Her prose is paraphrased. The term "on-shell" is not hers. See Sources.