Noether's 1918 left-open group-theoretic questions: closure of variational symmetry groups containing derivatives of $u$, recovery of infinite groups from infinitesimal transformations, and the most general improper-energy groups

In her 1918 paper “Invariant Variational Problems”, Emmy Noether proved her two theorems relating the variational symmetry groups of a functional to its conservation laws and differential identities, together with their converses, but explicitly left several group-theoretic questions undecided. This record audits those questions as one research problem: whether the infinitesimal transformations of an admitted group still form a group after the trivial ones (those with $\bar\delta u = 0$ and $\mathrm{Div}(f\cdot\Delta x)=0$) are eliminated; whether the exceptional case in her converse proofs, in which derivatives of $u$ occur in $\Delta x,\Delta u$ and the group property is restored only by adjoining all $\Delta x$ with $\mathrm{Div}(f\cdot\Delta x)=0$, can actually arise; whether the finite transformations of an infinite continuous group are always recoverable from infinitesimal transformations whose arbitrary functions are independent of the integration parameter (Lie's question about analyticity in the arbitrary functions); and what the most general infinite group containing the displacement group is that forces the energy relations of an invariant integral to be improper.

Background

Emmy Noether's 1918 paper “Invariant Variational Problems” studies variational integrals

$$I = \int f\big(x, u, u_x, u_{xx}, \ldots\big)\,dx$$

that are invariant under a continuous group of transformations in the sense of Sophus Lie. Here $x = (x_1, \ldots, x_n)$ are the independent variables, $u = (u^1, \ldots, u^q)$ are the dependent variables (fields), $f$ is the Lagrangian, $u_x$, $u_{xx}$, and so on denote its derivatives, and $dx$ is the volume element. A finite continuous group $G_\rho$ depends on $\rho$ essential parameters, while an infinite continuous group $G_{\infty\rho}$ depends on $\rho$ arbitrary functions $p^{(\lambda)}(x)$ and their derivatives up to a finite order $\sigma$. An infinitesimal transformation of such a group is written $x \mapsto x + \Delta x$, $u \mapsto u + \Delta u$, and Noether allows $\Delta x$ and $\Delta u$ to depend not only on $x$ and $u$ but also on derivatives of $u$; these derivative-dependent generators are exactly the objects rediscovered in the 1960s and now called generalized, higher-order, or Lie–Bäcklund symmetries. She computes with the associated variation, today called the evolutionary representative,

$$\bar\delta u^i = \Delta u^i - \sum_\lambda \frac{\partial u^i}{\partial x_\lambda}\,\Delta x_\lambda,$$

and with the infinitesimal invariance condition $\bar\delta f + \mathrm{Div}(f \cdot \Delta x) = 0$, where $\mathrm{Div}$ is the total divergence. The Euler–Lagrange expressions, her “Lagrangian expressions” $\psi_i$, are the components of the variational derivative of $f$, so the first variation of $I$ is $\delta I = \int \sum_i \psi_i\,\bar\delta u^i\,dx$ up to boundary terms.

Noether's First Theorem states that invariance of $I$ under a $G_\rho$ yields $\rho$ linearly independent combinations of the $\psi_i$ that become divergences, i.e. conservation laws, and, conversely, that $\rho$ such divergence relationships imply invariance of $I$ under a $G_\rho$; her Second Theorem states that invariance under a $G_{\infty\rho}$ yields $\rho$ differential identities, called dependencies, among the $\psi_i$ and their derivatives, and conversely. She regarded the converses as the point of the paper, writing later in a referee report that she had set out above all “to state the converse” (quoted by Pitts 2016). The proofs run into infinitesimal transformations that are trivial in the sense that $\bar\delta u = 0$ and $\mathrm{Div}(f \cdot \Delta x) = 0$, which can occur only when $\Delta x, \Delta u$ contain derivatives of $u$; these are to be eliminated from the group, with only the remaining parameters or arbitrary functions counted, while whether the remaining infinitesimal transformations still form a group “must be left moot” (§2, note 13). In the converse for finite groups (§3) she proves that once the transformations are free of derivatives of $u$ the finite transformations generated by integration always form a group; if derivatives occur, the group property may have to be restored by adjoining all $\Delta x$ with $\mathrm{Div}(f\cdot\Delta x) = 0$, whose parameters by agreement do not count, and whether this case can arise is left undecided. For infinite groups (§4) the same adjunction of trivial transformations may be necessary before the conclusions can be drawn. There she integrates the infinitesimal transformations to recover the finite ones, asserting that the group generated from the infinitesimal transformations of a $G_{\infty\rho}$ reduces back to $G_{\infty\rho}$ (note 18), while the question whether it always suffices to take the arbitrary functions independent of the integration parameter $t$, the case in which the dependency on the arbitrary function becomes analytic, was raised in another form by Lie and is left open (note 19). Finally, in §6 she proves in generalized group-theoretic form an assertion Hilbert had made about general relativity: if $I$ admits the displacement group (the translations $\Delta x_i = \epsilon_i$, $\Delta u = 0$, with $\bar\delta u^i = -\sum_\lambda \epsilon_\lambda\,\partial u^i/\partial x_\lambda$), then the associated energy relations become improper — in modern language the conservation laws are trivial, vanishing identically rather than on solutions of the field equations — if and only if $I$ is invariant under an infinite group containing the displacement group as a subgroup. Her examples of such groups are the induced coefficient transformations of total differential forms, a logarithmic example, and groups obtained from $I$ by adding a boundary integral; whether these exhaust the most general such groups “must remain undecided”.

The modern theory relocates these questions rather than answering them verbatim. A generalized symmetry is an evolutionary vector field on the infinite jet space — the inverse limit of the finite jet spaces recording $x$, $u$ and all their derivatives — whose characteristic $Q[u]$ is a differential function; the infinitesimal symmetries of a differential equation form a Lie algebra under the commutator bracket, the infinitesimal analogue of the group property (Olver 2018; Olver 2021; systematized in Olver 1993). The trivial transformations are handled by equivalence relations: a divergence symmetry, meaning an infinitesimal transformation leaving the integrand invariant up to a divergence term, is trivial if its evolutionary representative vanishes on solutions or its divergence vanishes identically; conservation laws trivial of the first kind (the current itself vanishes on solutions) and of the second kind (the current is itself a divergence) are defined analogously; and for Lagrangians whose Euler–Lagrange systems are normal, meaning they can be put in Cauchy–Kovalevskaya form, i.e. solved for the highest derivatives, the Noether correspondence induces a one-to-one map between equivalence classes of divergence symmetries modulo trivial ones and equivalence classes of conservation laws modulo trivial ones, the result Kosmann-Schwarzbach (2020) calls the Noether–Olver–Vinogradov theorem. Olver (2018) proves the refined converse of the first theorem in this form and flags an adjacent open problem, whether any overdetermined systems match trivial symmetries with nontrivial conservation laws; Olver (2021) proves a modern converse of the second theorem: every underdetermined system of Euler–Lagrange equations, meaning one satisfying a nontrivial differential identity, admits an infinite hierarchy of generalized symmetries depending on arbitrary differential functions, and every linear dependency among the Euler–Lagrange equations produces an infinite-dimensional variational symmetry algebra; related statements hold in the Batalin–Vilkovisky formalism, a homological quantization framework for gauge theories (Fulp, Lada, and Stasheff 2003). These results settle the infinitesimal, modular core of the closure questions for normal Euler–Lagrange systems, but they do not answer Noether's literal questions about finite transformation groups of $(x, u)$ or her literal §2 closure question outside that regime; as she observed at the end of §3, when $\Delta x$ and $\Delta u$ contain derivatives of $u$ the finite transformations may depend on infinitely many derivatives of $u$, so they no longer act on $(x, u)$ in the ordinary sense. The structure theory of Lie pseudogroups, infinite transformation groups reconstructed from infinitesimal data, develops related machinery without returning to note 19 (Olver, Pohjanpelto, and Valiquette 2009).

On the physics side, Noether's biconditional on improper energy, often called the converse Hilbertian assertion, has had an active afterlife: Pitts (2016) documents how little attention her converses received and shows that the particle-physics derivation of Einstein's equations from a massless spin-2 field reinvented the converse Hilbertian assertion; reviews of gravitational energy use it (Petrov and Pitts 2020; Deser 2019, who stresses that breaking coordinate invariance through boundary conditions removes the barrier the second theorem otherwise poses to ordinary energy conservation); De Haro (2021) reviews it and refines how the translation generators must sit inside the infinite group, Maxwell theory showing that the naive reading is false. Classification results in the same spirit exist only for specific theories: Anderson and Torre (1996) classified all local generalized symmetries of the vacuum Einstein equations, finding only infinitesimal generalized diffeomorphisms and constant metric scalings, with no nontrivial conservation laws attached. The most general improper-energy-forcing groups, Lie's analyticity question, and Noether's literal closure questions have not been settled in the literature.

Problem Statement

Determine the group-theoretic status of the transformation groups in Noether's two theorems and their converses by resolving the questions her 1918 paper explicitly leaves undecided — in her own setting of continuous groups of finite transformations of $(x, u)$ whose infinitesimal transformations $\Delta x, \Delta u$ may contain derivatives of $u$.

1. Closure after eliminating trivial transformations (§2, note 13). The trivial infinitesimal transformations, those with $\bar\delta u = 0$ and $\mathrm{Div}(f \cdot \Delta x) = 0$, are always to be eliminated from an admitted group, and only the remaining parameters or arbitrary functions counted. This is the literal §2 question in Noether's own setting; since that setting is not restricted to normal Euler–Lagrange systems, the modern equivalence-class correspondence for normal systems does not answer it verbatim. Do the remaining infinitesimal transformations still form a group in every case, or is there an invariant variational problem for which the elimination destroys the group property?

2. Existence of the exceptional augmentation case (§3, and §4). In the converse proofs, if derivatives of $u$ occur in $\Delta x$ or $\Delta u$, the transformations obtained may fail to have the group property unless $\Delta x$ is augmented by all functions with $\mathrm{Div}(f \cdot \Delta x) = 0$, whose parameters by agreement do not count; the mechanism she envisions is that the bracket of two of the obtained infinitesimal transformations falls into the trivial class instead of being a linear combination of the original ones. Does such a case actually arise — is there a concrete Lagrangian $f$ and admitted group for which the augmentation (or, in §4, the adjunction of trivial transformations before drawing the conclusions) is genuinely necessary — or can it be proved that the case never arises?

3. Lie's question (§4, notes 18–19). Noether asserts that the group generated from the infinitesimal transformations of a $G_{\infty\rho}$ reduces back to $G_{\infty\rho}$, and that if it suffices to assume the arbitrary functions $p(x)$ free of the integration parameter $t$, the dependency on $q(x) = t\,p(x)$ becomes analytic. Decide whether this always occurs: is every finite transformation of an infinite continuous group of a variational problem obtained as the time-1 map of the flow of a single infinitesimal generator with parameter-independent arbitrary functions, or does there exist such a group for which the $t$-dependent integration is genuinely needed?

4. The most general improper-energy groups (§6). Characterize the most general infinite groups containing the displacement group as a subgroup that force the energy relations of an invariant integral $I$ to be improper. Noether's candidate classes are the induced transformations of the coefficients of total differential forms (in which only derivatives of the arbitrary functions occur), the boundary-integral variants of $I$, and special examples such as the logarithmic one; decide whether these exhaust all possibilities, and, if they do not, classify the remaining groups. Deciding exhaustion and completing the classification are each separately resolvable.

Questions 1–4 are independently resolvable: deciding any one of them is a partial advance, and the task is fully resolved only when all four are decided. The four questions jointly constitute the unfinished group-theoretic foundation of the converses of Noether's theorems.

References

1. Emmy Noether (1971), “Invariant Variation Problems”, English translation by M. A. Tavel of “Invariante Variationsprobleme”, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse (1918), 235–257; Transport Theory and Statistical Physics 1(3), 186–207; DOI 10.1080/00411457108231446, https://doi.org/10.1080/00411457108231446; arXiv:physics/0503066, https://arxiv.org/abs/physics/0503066
2. Emmy Noether (2011), “Invariant Variational Problems”, in Yvette Kosmann-Schwarzbach, The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, pp. 3–22; DOI 10.1007/978-0-387-87868-3_1, https://doi.org/10.1007/978-0-387-87868-3_1
3. Yvette Kosmann-Schwarzbach (2020), “The Noether theorems in context”, arXiv:2004.09254, https://arxiv.org/abs/2004.09254
4. Peter J. Olver (1993), Applications of Lie Groups to Differential Equations, second edition, Graduate Texts in Mathematics 107, Springer, New York; DOI 10.1007/978-1-4612-4350-2, https://doi.org/10.1007/978-1-4612-4350-2
5. Peter J. Olver (2018), “Emmy Noether's enduring legacy in symmetry”, Symmetry: Culture and Science 29(4), 475–485; DOI 10.26830/symmetry_2018_4_475, https://doi.org/10.26830/symmetry_2018_4_475; author's manuscript at https://www-users.cse.umn.edu/~olver/s_/noether.pdf
6. Peter J. Olver (2021), “Higher-order symmetries of underdetermined systems of partial differential equations and Noether's Second Theorem”, Studies in Applied Mathematics 147(3), 904–913; DOI 10.1111/sapm.12404, https://doi.org/10.1111/sapm.12404 (author's manuscript at https://www-users.cse.umn.edu/~olver/s_/under.pdf)
7. J. Brian Pitts (2016), “Einstein's Equations for Spin 2 Mass 0 from Noether's Converse Hilbertian Assertion”, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 56, 60–69; DOI 10.1016/j.shpsb.2016.08.008; arXiv:1611.02673, https://arxiv.org/abs/1611.02673
8. Alexander N. Petrov and J. Brian Pitts (2020), “The Field-Theoretic Approach in General Relativity and Other Metric Theories. A Review”, Space, Time and Fundamental Interactions 2019(4), 66–124; arXiv:2004.10525, https://arxiv.org/abs/2004.10525
9. Stanley Deser (2019), “Energy in Gravitation and Noether's Theorems”, Journal of Physics A: Mathematical and Theoretical 52(38), 381001; DOI 10.1088/1751-8121/ab1e19; arXiv:1902.05105, https://arxiv.org/abs/1902.05105
10. Sebastian De Haro (2021), “Noether's Theorems and Energy in General Relativity”, arXiv:2103.17160, https://arxiv.org/abs/2103.17160; published as a chapter in The Philosophy and Physics of Noether's Theorems, Cambridge University Press (2022), pp. 197–256, DOI 10.1017/9781108665445.010, https://doi.org/10.1017/9781108665445.010
11. Ian M. Anderson and Charles G. Torre (1996), “Classification of local generalized symmetries for the vacuum Einstein equations”, Communications in Mathematical Physics 176(3), 479–539; DOI 10.1007/BF02099248; arXiv:gr-qc/9404030, https://arxiv.org/abs/gr-qc/9404030
12. Peter J. Olver, Juha Pohjanpelto, and Francis Valiquette (2009), “On the Structure of Lie Pseudo-Groups”, SIGMA — Symmetry, Integrability and Geometry: Methods and Applications 5, 077; DOI 10.3842/SIGMA.2009.077; arXiv:0907.4086, https://arxiv.org/abs/0907.4086
13. Ronald O. Fulp, Thomas J. Lada, and James D. Stasheff (2003), “Noether's variational theorem II and the BV formalism”, in Proceedings of the 22nd Winter School “Geometry and Physics” (Srní, 2002), Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento 71, 115–126; arXiv:math/0204079, https://arxiv.org/abs/math/0204079