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{{DISPLAYTITLE:Bessel-Hagen and the first uses}}

Three years after Noether's paper, a young Göttingen mathematician used it, and
in using it added something she had not written. The addition is now attributed
to her more often than to him.

== Bessel-Hagen ==

Erich Bessel-Hagen was born in 1898 and died in 1946. He came to Göttingen and
fell under Klein's influence to an unusual degree: he lived in Klein's house and
served as his private assistant from 1921 to 1923, and later helped prepare
Klein's lectures on nineteenth-century mathematics for publication.

In September 1921 the ''Mathematische Annalen'' published his ''Über die
Erhaltungssätze der Elektrodynamik'' at pages 258 to 276 of volume 84. It is the
first substantial application of Noether's theorems, and his citations to her
are precise — to particular pages and particular sections of her paper, which he
abbreviates simply by her name.

== What the paper does ==

Bessel-Hagen applies both theorems to Maxwell's equations and asks what the full
symmetry group of electrodynamics yields.

The answer is larger than the Lorentz group. Electrodynamics is invariant under
the fifteen-parameter conformal group, a fact established by Bateman and
Cunningham a decade earlier and known then as the group of spherical wave
transformations. Feeding that group into the first theorem produces conserved
quantities that had not been written down, beyond the familiar energy, momentum
and angular momentum.

== The addition: invariance up to a divergence ==

Noether's theorems as she stated them require the action integral to be
invariant. Bessel-Hagen observed that this is stronger than necessary. It is
enough that the integrand change by a divergence, since a divergence integrates
to a boundary term and leaves the variational problem alone.

This is now called divergence invariance, or quasi-invariance, and the
symmetries it admits are called divergence symmetries. Bessel-Hagen also
identified why the weaker condition has to be defined carefully. Strict
invariance under an infinitesimal transformation carries over automatically to
the one-parameter group that transformation generates. Invariance up to a
divergence does not. The definition therefore has to be attached to the
infinitesimal transformations, not to the finite group — a point that is
routinely skipped in modern statements.

The standard illustration is the Galilean boost. The Lagrangian of a free
particle is not invariant under a boost; it changes by a total time derivative.
Under strict invariance the boost gives nothing. Under divergence invariance it
gives a conserved quantity, and that quantity is the uniform motion of the
centre of mass.

'''This is a correction to the usual account, and this volume marks it
everywhere it arises.''' Divergence symmetries are commonly presented as part of
Noether's theorem. They are not in her paper. She works with invariance of the
integral, relation (1) of her first section. She does observe that an integral
differing from another by a divergence admits the same group with the same
variation, which is adjacent to Bessel-Hagen's point and is not the same as it.
The generalisation is his.

== The first uses, and their scale ==

There were few. A study of Bessel-Hagen's paper reports that it drew its first
proper citation in 1934 and only a handful more over the following two decades.

That figure describes his paper. It is not evidence about the reception of
Noether's, and this volume does not treat it as such. What the reception of
Noether's paper actually was over the same period is
[[Noether's theorems/History/The long neglect|unresolved]].

''Provenance: Bessel-Hagen's paper was not reached in full text. It is indexed
and digitised, but no copy was retrievable in the course of preparing this
volume, and the only English translation is under copyright. Nothing above is
quoted. The account of the paper's contents and of its citation history is taken
from the secondary literature. Closing this gap requires the ''Mathematische
Annalen'' volume 84 scan. See [[Noether's theorems/Sources|Sources]].''

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