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Noether theorems volume, initial import (H-05)

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{{DISPLAYTITLE:The precursors}}

Noether opens her paper by naming the people who had done parts of this before
her. That list is the best evidence available about what she thought she was
adding, and it is worth reading before anyone else's account of the same
question.

== Whom she named ==

Her second sentence says that the combination of methods she is using — the
formal calculus of variations joined to Lie's theory of groups — is not new for
special cases. She then names five parties, sorted by which kind of group they
had handled.

For special '''finite''' groups, meaning groups with a fixed number of
parameters: '''Hamel''' and '''Herglotz'''.

For special '''infinite''' groups, meaning groups depending on arbitrary
functions: '''Lorentz and his pupils''', with Fokker named as the example, and
'''Weyl''' and '''Klein'''.

She adds in a footnote that a paper by '''Kneser''' had just appeared, dealing
with the construction of invariants by a similar method. Elsewhere in the paper
she cites '''Engel'''. And the whole apparatus of continuous groups is
'''Lie's''', cited by name in her third footnote to a specific work and later
to a specific theorem inside it.

== The loci she gives ==

Her footnotes carry the references, and they are short. Hamel is cited to
''Mathematische Annalen'' volume 59 and to the ''Zeitschrift für Mathematik und
Physik'' volume 50. Herglotz is cited to ''Annalen der Physik'', fourth series,
volume 36, with section 9 at page 511 singled out. Fokker is cited to the
''Verslag'' of the Amsterdam Academy, 27 January 1917. Kneser is cited to
''Mathematische Zeitschrift'' volume 2. Engel is cited to the Göttingen
''Nachrichten'' for 1916 at page 270. For further literature she refers the
reader to Klein's second note rather than compiling it herself.

Lie is cited to ''Grundlagen für die Theorie der unendlichen kontinuierlichen
Transformationsgruppen'', in the reports of the Saxon Academy for 1891, which
she abbreviates as ''Grundlagen'' throughout. She uses it three times: for the
definition of an infinite continuous group as one whose transformations are the
general solutions of a system of partial differential equations; for the
normalisation that puts the identity at zero parameter values; and, at the
decisive point of her fourth section, for Theorem VII at page 391, which is what
lets her pass from a group of infinitesimal transformations back to a group of
finite ones.

'''A caution on Hamel.''' Secondary reference lists in this area sometimes give
Hamel's relevant papers as ''Mathematische Annalen'' volume 66 and a 1917
''Jahresbericht'' note. Those are not the loci Noether gives. Either she is
pointing at different papers or one of the two lists is wrong. This has not been
checked and the citations above are hers, read from her own footnotes.

== What is not settled ==

Naming a precursor is not the same as measuring what they anticipated, and
Noether does neither. Her sentence says the combination of methods is not new in
special cases. It does not say how much of either theorem was already in Hamel,
or in Herglotz, or in Lie.

This volume does not answer that question. Answering it means reading the eight
cited papers against the two theorems, and deciding for each what it establishes
and under what restriction — work that has not been done here.

What can be said is where the boundary of the claim sits. Both theorems are
statements at a generality nobody in the list attempted: arbitrary numbers of
independent variables, derivatives of arbitrary order, transformations mixing
dependent and independent variables, and both converses. Anticipation of a
special case is not anticipation of that. But this volume states that as the
shape of the question, not as its answer.

''Provenance: the names, the loci and the finite-against-infinite sorting are
read directly from the German original, which is public domain. None of the
cited precursor papers was read. No priority claim of any kind is made or
implied by the ordering above. See [[Noether's theorems/Sources|Sources]].''

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