The converse of the second theorem

Plain-language counterpart: 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, linearity in the parameters was justified as a corollary of the converse. Here it follows more directly, from the fact that ${𝔊}_{\mathrm{\infty }\varrho }$ depends formally on $\varrho$ and only $\varrho$ 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 $p$ and its derivatives. Compose two such transformations, one in $p$ and one in $q$, and collect the terms of lowest order. If any coefficient beyond the first two is non-zero, a term of the form $p^{\nu -\sigma }(\partial p/\partial x{)}^{\sigma }+q^{\nu -\sigma }(\partial q/\partial x{)}^{\sigma }$ occurs for some $\sigma >1$, 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 $p^{(\lambda )}(x)$ and add. The identical transformation (N14) then gives

$\sum {\psi }_i\overline{\delta }{u}_i=\mathrm{D}\mathrm{i}\mathrm{v}\Gamma \mathrm{(}\mathrm{1}\mathrm{)}$

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

That they form a group follows as before: otherwise composition would produce more arbitrary functions, while by assumption there are only $\varrho$ 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 $𝔊$ of finite transformations thereby defined. Each finite transformation is generated from infinitesimal ones, and so arises by integrating the simultaneous system

$\frac{d{x}_i}{dt}=\Delta x_i;\frac{d{u}_i}{dt}=\Delta u_i(\begin{array}{c}{x}_i=y_i\\ u_i=v\end{array}\text{for}t=0)\mathrm{(}\mathrm{2}\mathrm{)}$

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

Where derivatives of the $u$ occur, it may be necessary to adjoin the infinitesimal transformations with $\overline{\delta }u=0$ and $\mathrm{D}\mathrm{i}\mathrm{v}(f\cdot \Delta x)=0$ before the same conclusions can be drawn — the same qualification that 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 $x$, not on the $u$ and their derivatives.

If they did depend on the $u$, the identical transformation (N14), and so (N15), would contain derivatives of $p$ with respect to $u$ as well. Taking the $p^{(\lambda )}$ successively of zeroth, first and higher degree in the $u$ and their derivatives, with arbitrary functions of $x$ 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 Sources.