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{{DISPLAYTITLE:The converse of the second theorem}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Reading the second theorem backwards|Reading the second theorem backwards]].''

Her fourth section. It has two parts: a preliminary that discharges the linearity
assumption, and the converse proper.

== Linearity, discharged without the converse ==

For the finite group,
[[Noether's theorems/First theorem/The converse of the first theorem|linearity in
the parameters]] was justified as a corollary of the converse. Here it follows
more directly, from the fact that
<math>\mathfrak{G}_{\infty\varrho}</math> depends formally on
<math>\varrho</math> and only <math>\varrho</math> arbitrary functions.

The argument is a counting argument on composition. Suppose the dependence were
nonlinear, so that the transformation contains terms in powers of the
<math>p</math> and its derivatives. Compose two such transformations, one in
<math>p</math> and one in <math>q</math>, and collect the terms of lowest order.
If any coefficient beyond the first two is non-zero, a term of the form
<math>p^{\nu-\sigma}(\partial p/\partial x)^{\sigma}+q^{\nu-\sigma}(\partial q/\partial x)^{\sigma}</math>
occurs for some <math>\sigma>1</math>, and such a term cannot be written as a
differential quotient of a single function, nor as a power product of one. The
number of arbitrary functions has therefore increased, contrary to assumption.
If all those coefficients vanish, the second term is the differential quotient of
the first and linearity holds after all.

So the infinitesimal transformations satisfy a system of linear partial
differential equations, and since the group property holds they form what Lie
calls an infinite group of infinitesimal transformations.

== The converse ==

Given the dependencies (N16), multiply by <math>p^{(\lambda)}(x)</math> and add.
The identical transformation (N14) then gives

<math>\sum\psi_{i}\bar{\delta}u_{i}=\mathrm{Div}\,\Gamma\qquad\qquad\mathrm{(1)}</math>

and from this, exactly as in the finite case, <math>\Delta x</math> and
<math>\Delta u</math> are determined and the invariance of <math>I</math> under
those infinitesimal transformations follows. They depend linearly on
<math>\varrho</math> arbitrary functions and their derivatives up to the
<math>\sigma</math>th order.

That they form a group follows as before: otherwise composition would produce
more arbitrary functions, while by assumption there are only
<math>\varrho</math> dependencies (N16).

== From infinitesimal back to finite ==

This is where Lie is doing load-bearing work rather than supplying vocabulary.

An infinite group of infinitesimal transformations consists, by Lie's Theorem VII
at page 391 of the ''Grundlagen'', of the most general infinitesimal
transformations of a certain infinite group <math>\mathfrak{G}</math> of finite
transformations thereby defined. Each finite transformation is generated from
infinitesimal ones, and so arises by integrating the simultaneous system

<math>\frac{dx_{i}}{dt}=\Delta x_{i};\qquad\frac{du_{i}}{dt}=\Delta u_{i}\qquad\left(\begin{array}{c}x_{i}=y_{i}\\u_{i}=v\end{array}\ \text{for}\ t=0\right)\qquad\qquad\mathrm{(2)}</math>

with the qualification that it may be necessary to let the arbitrary
<math>p(x)</math> depend on <math>t</math> as well. If it suffices to take them
free of <math>t</math>, the dependence is analytic in the arbitrary functions
<math>q(x)=t\cdot p(x)</math>.

Where derivatives of the <math>u</math> occur, it may be necessary to adjoin the
infinitesimal transformations with <math>\bar{\delta}u=0</math> and
<math>\mathrm{Div}(f\cdot\Delta x)=0</math> before the same conclusions can be
drawn — the same qualification that
[[Noether's theorems/First theorem/The converse of the first theorem|the finite
converse]] carries.

== Two restrictions that turn out not to be restrictions ==

Noether closes the section by removing an assumption made silently earlier: that
the arbitrary functions depend only on the <math>x</math>, not on the
<math>u</math> and their derivatives.

If they did depend on the <math>u</math>, the identical transformation (N14), and
so (N15), would contain derivatives of <math>p</math> with respect to
<math>u</math> as well. Taking the <math>p^{(\lambda)}</math> successively of
zeroth, first and higher degree in the <math>u</math> and their derivatives, with
arbitrary functions of <math>x</math> as coefficients, gives dependencies (N16)
again, only more of them — and by the converse just proved, these reduce to the
earlier case.

The same argument shows that simultaneous dependencies and independent divergence
relations correspond to mixed groups, which closes the last gap in the
classification.

''Provenance: the whole argument is Noether's fourth section, and the Lie theorem
is her citation, not read independently. Display (2) is her simultaneous system,
matching (N17). Reproduced in her notation from the public-domain German
original; her prose is paraphrased. See
[[Noether's theorems/Sources|Sources]].''

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