Noether's theorems/First theorem/One-parameter groups and their generators

Plain-language counterpart: What a continuous symmetry is.

A one-parameter group: flow lines and the generator

The two theorems differ in exactly one input: the kind of group. This page defines both kinds, because the distinction has to be in place before either theorem can be stated.

Group, finite, infinite, mixed

A transformation group is a system of transformations such that every transformation has an inverse in the system, and the composition of any two belongs to the system.

The group is called finite continuous, written ${𝔊}_{\varrho }$, when its transformations are contained in a most general one depending analytically on $\varrho$ essential parameters $\epsilon$. Essential means the $\varrho$ parameters cannot be represented as $\varrho$ functions of fewer parameters.

The group is called infinite continuous, written ${𝔊}_{\mathrm{\infty }\varrho }$, when its most general transformations depend on $\varrho$ essential arbitrary functions $p(x)$ and their derivatives, analytically or at least continuously and finitely often continuously differentiably.

Between the two sits the group depending on infinitely many parameters but not on arbitrary functions. A group depending on both arbitrary functions and parameters is called mixed.

Noether takes her definition of the infinite continuous group from Lie, who defines it as a transformation group whose transformations are given by the most general solutions of a system of partial differential equations, provided those solutions do not depend on only a finite number of parameters.

The first theorem takes ${𝔊}_{\varrho }$. The second takes ${𝔊}_{\mathrm{\infty }\varrho }$. That is the whole of the difference in input, and everything that follows from it is downstream of this one distinction.

Normalising the identity

For any continuous group, finite or infinite, the parameters can always be arranged so that the identity transformation corresponds to zero values of the $\epsilon$, or of the arbitrary functions $p(x)$. Noether cites Lie for this. Where arbitrary functions are involved, the special parameter values are replaced by fixed functions and their derivatives.

The general transformation and its lowest terms

With the identity at zero, the most general transformation has the form

$\begin{array}{ccc}{y}_i& =A_i(x,u,\frac{\partial u}{\partial x},\dots )& =x_i+\Delta x_i+\dots \\ v_i(y)& =B_i(x,u,\frac{\partial u}{\partial x},\dots )& =u_i+\Delta u_i+\dots \end{array}\mathrm{(}\mathrm{1}\mathrm{)}$

where $\Delta x_i$ and $\Delta u_i$ are the terms of lowest dimension in $\epsilon$, or in $p(x)$ and its derivatives. Noether assumes them linear in those quantities and remarks that this will be shown to be no restriction of generality — the demonstration is in the converse.

The infinitesimal transformation is what is left when the higher terms are dropped:

$y_i=x_i+\Delta x_i;v_i(y)=u_i+\Delta u_i\mathrm{(}\mathrm{2}\mathrm{)}$

Recovering the finite transformations

The infinitesimal transformations are not a weaker object. The finite ones are obtained back from them by integrating the simultaneous system

$\frac{d{x}_i}{dt}=\Delta x_i;\frac{d{u}_i}{dt}=\Delta u_i;(\begin{array}{c}{x}_i=y_i\\ u_i=v_i\end{array}\text{for}t=0)\mathrm{(}\mathrm{N}\mathrm{17}\mathrm{)}$

and the resulting finite transformations contain $\varrho$ parameters, namely the products $t{\epsilon }_1,\dots ,t{\epsilon }_{\varrho }$.

This is why the whole of both proofs can be run on infinitesimal transformations without loss: the group is recoverable from its generators. Noether uses the recovery explicitly in both converses.

One caution. When $\Delta x$ and $\Delta u$ contain derivatives of the $u$, the finite transformations obtained this way can depend on infinitely many derivatives, because the count grows at each step of the integration. That is treated on the prolongation page.

Provenance: (N17) is Noether's equation (17), reproduced in her notation from the public-domain German original. Equations (1) and (2) are displays this volume has numbered; the array in (1) is hers, set unnumbered in the original. Her prose is paraphrased and the Lie citations are hers, not read independently. See Sources.