View source for Noether's theorems/Second theorem/The two theorems side by side

{{DISPLAYTITLE:The two theorems side by side}}

''Plain-language counterpart: [[Noether's theorems/Explainers/The two theorems compared|The two theorems compared]].''

[[File:NT-F-06.png|thumb|right|360px|alt=A three-column comparison table. Rows: what each theorem takes, what it returns, where it holds, what happens when the field equations are imposed, whether a conserved quantity results, and whether the converse is proved. Theorem II's conserved-quantity row reads none.|The first and second theorems side by side]]

Every page in this volume that needs the distinction links here. The comparison
below is read off the two finished proofs, not asserted.

== The table ==

{| class="wikitable"
! !! Theorem I !! Theorem II
|-
! Group
| finite continuous, <math>\varrho</math> constant essential parameters
| infinite continuous, <math>\varrho</math> arbitrary functions, entering to order <math>\sigma</math>
|-
! Output
| <math>\varrho</math> linearly independent combinations of the Lagrange expressions become divergences
| <math>\varrho</math> identical relations among the Lagrange expressions and their derivatives to order <math>\sigma</math>
|-
! Form of the output
| <math>\sum\psi_{i}\bar{\delta}u_{i}^{(\lambda)}=\mathrm{Div}\,B^{(\lambda)}</math>
| left side of (N16) <math>=0</math>
|-
! Where it holds
| identically, in all arguments
| identically, in all arguments
|-
! On setting <math>\psi_{i}=0</math>
| becomes <math>\mathrm{Div}\,B^{(\lambda)}=0</math>, a conservation law
| becomes the statement that <math>\varrho</math> field equations follow from the rest
|-
! Conserved quantity
| yes: the current <math>B^{(\lambda)}</math>, the charge its cross-section
| none. Nothing flows
|-
! Converse
| proved, section 3, with one case left undecided
| proved, section 4
|-
! Where the proofs part
| (N12) splits by equating coefficients of the <math>\varepsilon</math>
| (N12) cannot split; derivatives of <math>p</math> are removed by (2) of S-02, then the boundary choice kills the divergence
|}

== The one place they diverge ==

Both proofs run identically from (N1) through (N9), (N11), (N3) and into

<math>\sum\psi_{i}\bar{\delta}u_{i}=\mathrm{Div}\,B\qquad\qquad\mathrm{(N12)}</math>

That is roughly nine tenths of the work, and it is shared. What follows depends
on one property of <math>\bar{\delta}u</math>.

If <math>\bar{\delta}u</math> is linear in constant parameters, the parameters
are independent, (N12) holds coefficient by coefficient, and
<math>\varrho</math> divergence relations drop out.

If <math>\bar{\delta}u</math> is linear in arbitrary functions '''and their
derivatives''', no such split exists, because <math>\partial p/\partial x</math>
is not independent of <math>p</math>. The derivatives must first be traded away
against the coefficients, which generates a divergence; then the freedom to
choose <math>p</math> vanishing on the boundary kills the whole right-hand side,
and what is left is zero rather than a divergence.

'''The right-hand side is the difference.''' One theorem ends with a divergence
on the right, the other with nothing. Everything else follows from that.

== Three errors this page exists to prevent ==

'''"The second theorem is the local version of the first."''' It is not a version
of the first. Different input, different output, different consequence of
imposing the field equations. The table's third and sixth rows are the check.

'''"The second theorem gives conservation laws for local symmetries."''' It gives
identities. When a local symmetry does produce something that looks like a
conservation law, that law is improper in the sense of
[[Noether's theorems/Second theorem/Proper and improper conservation laws|the
criterion page]] — its conserved quantity is assembled from the field equations
and carries no information they did not.

'''"General relativity is why energy conservation is degenerate."''' The
criterion is group-theoretic and biconditional. General relativity satisfies its
condition. So does the logarithmic group Noether constructs with nothing physical
attached. Her closing remark says this in as many words.

== Which one is the paper about ==

Both, but not equally. Theorem I is stated first and is the one that entered
physics. Theorem II is the one the paper was written to prove: it is the answer to
the Göttingen question, it occupies the final section, and it is what
Hilbert's assertion needed.

This volume's editorial position, stated as such: '''quoting only Theorem I gives
the corollary without the argument.''' Readers are free to disagree, and
[[Noether's theorems/History/What Noether's theorem now means|the history track]]
records how the other reading came to dominate.

''Provenance: every row of the table is read off the two proofs in this volume,
which are in turn read from the public-domain German original. The three errors
and the closing position are this volume's own and are marked as editorial. See
[[Noether's theorems/Sources|Sources]].''