In 1918, the mathematician Emmy Noether published two wonderful theorems that had a tremendous impact in physics, mathematics, and beyond. While Noether's primary interest and lasting contribution to mathematics was laying the foundations of modern abstract algebra, the term “Noether's Theorem” belongs to the lexicon of physicists and applied mathematicians.

Nevertheless, many of them remain unaware of the true scope and formulation of her fundamental theorems. In part, this is due to the inadequate and misinformed treatments of her results that continue to proliferate in the literature. Unfortunately, despite the best of intentions, the book under review is of this very nature.

Noether's First Theorem—the one in the book's title—establishes a one-to-one correspondence between the continuous (Lie) symmetry groups of a variational principle and the conservation laws of the associated Euler-Lagrange equations, whose solutions are the (smooth) extrema (more correctly, stationary points). While Neuenschwander states that symmetries produce conservation laws, he does not mention the reverse, which is an integral part of her statement of the Theorem (see below). Noether's Second Theorem states that if the variational principle admits an infinite-dimensional symmetry group depending on one or more arbitrary functions of the physical coordinates, then the associated conservation laws are trivial, but there are nontrivial differential identities among the field equations, which thus form an underdetermined system of differential equations. For proofs of both results see Ref. 1.

Neuenschwander's book does a commendable job detailing Noether's personal and professional history, illustrated by numerous quotes, and how she came to these theorems. In brief, in 1915, Noether, as a leading young expert (albeit unpaid due to her sex) in invariant theory and Lie groups, was invited by David Hilbert and Felix Klein to visit the University of Göttingen. The reason for Hilbert's invitation was that so she could help him in his intense ongoing competition with Einstein to establish the foundations of general relativity and, in particular, to resolve an apparent paradox: the triviality of the energy conservation law derived from time-translational symmetry of the Hilbert variational principle. (At that time, many special cases of Noether's Theorem, including the connections between translational and rotational symmetries and conservation of linear and angular momentum, and time translations with conservation of energy, were already known.^{1,2}) Noether's Second Theorem resolved Hilbert's dilemma; as she showed, the triviality of the energy conservation law was because the time translational symmetry group belongs to such an infinite-dimensional variational symmetry group, that, consequently, produces the Bianchi identities among the field equations.

Despite the fundamental importance of her theorems in classical and quantum field theories as well as in mathematical analysis, for the most part there have been major misunderstandings about what she actually accomplished in her seminal paper. In Ref. 1, I state and prove the full versions and further argue that her paper contains another fundamental but largely unrecognized contribution—the introduction and application of generalized symmetries, meaning those whose infinitesimal generators are allowed to depend upon the derivatives of the field variables—which did not appear in the earlier literature. A half century later, such higher order symmetries and their consequential higher order conservation laws played an essential role in the discovery of integrable (soliton) partial differential equations, such as the nonlinear Schrödinger and Korteweg-deVries equations. Their importance for Noether was that they allowed her to obtain the aforementioned one-to-one correspondence between symmetries and conservation laws. While writing Ref. 1, I started investigating the history of her Theorems in the literature, which contains a strange mixture of papers claiming special cases to be the “Noether Theorem,” followed by a rash of subsequent papers purporting to generalize it, when they were merely restating special cases of her marvelous and very general result. My initial historical forays were taken up in earnest by Yvette Kosmann-Schwarzbach in her masterly history of Noether's Theorem and its reception and development over the last century.^{2}

Unfortunately, Neuenschwander's book is representative of the aforementioned genre, stating a special case of her first result as if it were the general Noether Theorem. If he read Noether's original paper, he did not fully understand it. Nor did he consult the detailed discussion of the two Theorems in the first edition of Ref. 1. Even worse, in his revision Neuenschwander cites Kosmann-Schwarzbach's book, so there is no excuse for remaining ignorant of what is written in it. Indeed, a major concern is that his book will foster yet another generation of physicists who do not understand the full scope and power of Noether's First Theorem.

Neuenschwander also appears to be confused by Noether's Second Theorem. On Page 8 he says it includes the first as a special case, and repeats this claim on page 203, where he makes the bizarre claim that the Second Theorem is the Noether Identity. Now, while the Second Theorem does rely on this fundamental identity that underlies the First Theorem, this is not the point. The Second Theorem, as stated by Noether, only applies to certain kinds of infinite-dimensional symmetry groups, e.g., gauge symmetries, that depend upon arbitrary functions of the independent variables, whereas the First Theorem and its key identity apply to *all* continuous symmetry groups (both finite-dimensional Lie groups and infinite-dimensional Lie pseudo-groups), the only issue being triviality of the resulting conservation laws. The latter question was finally dealt with in Ref. 1, where it was shown that a “normal system” (meaning one without integrability conditions) has a one-to-one correspondence between nontrivial symmetry groups and nontrivial conservation laws. Underdetermined systems of Euler-Lagrange equations, such as those arising in general relativity, fall under the ambit of Noether's Second Theorem, and admit nontrivial differential relations among the field equations, such as the relativistic Bianchi identities. These points are properly explained in the relativistic framework on pages 225–234, but the initial characterization of the Second Theorem on pages 8 and 203 remains deeply flawed.

Neuenschwander also exhibits a rather shaky knowledge of the calculus of variations. His derivation of the Euler-Lagrange equations is unnecessarily complicated. In particular on pages 37–38, the variation $\u03c2$ is assumed to be continuously differentiable but the lemma used to complete the proof only assumes its continuity, and hence is not immediately applicable. It is also worth pointing out that only sufficiently smooth extrema satisfy the Euler-Lagrange equations. On page 33, he makes the strange claim that maxima and minima “lie outside the mathematics of the calculus of variations,” and later on page 94 states that “when a functional (sic) is said to be a minimum and not a maximum, or vice versa, it is for physical reasons, not mathematical ones.” This effectively ignores the entire history of the calculus of variations, particularly the theory of the second variation, the importance of conjugate points, the variety of necessary and sufficient extremal conditions due to Legendre, Weierstrass, Erdmann, Jacobi, Hilbert, Caratheodory, etc., none of which are mentioned in the text. And this does not even include powerful direct methods based on modern functional analysis. He also misstates a number of basic results in analysis. For example, page 36 claims that Leibniz's rule allows one to bring derivatives under the integral sign. But this rule is merely the formula for the derivative of the product of two functions and thus justifies integration by parts, which underlies all calculations in the classical calculus of variations, including Noether's Identities and Theorems.

The descriptions of symmetry and group theory are particularly poor. He gives a reasonable explanation of an infinitesimal transformation, but then, in Exercise 4.7 describes the infinitesimal generators in terms of matrices, which only works for linear actions and is false as stated (and the definition of “Killing vector” is not correct). He fails to develop the connection between the infinitesimal generator, which should be thought of as a vector field on the underlying space, and the induced one-parameter group, which can be identified with the flow of the vector field in the sense of dynamical systems. This is basic physics of, say, fluid mechanics, where the velocity vector field generates a steady state fluid flow, with time playing the role of the group parameter. Furthermore, I could not find a clear statement that the symmetries of the variational problem are symmetries of the Euler-Lagrange equations, but not conversely, the most common counterexamples being scaling groups. Page 79 says “functionals can be extremals but not invariant, and they can be invariant but not extremal” which makes no sense at least to me, in the same fashion as the above quoted sentence on page 94. How can a functional (as opposed to a solution) be extremal?

Sophus Lie, whose remarkable theory of symmetry groups of differential equations underlies Noether's results, makes only a cameo appearance on page 75. In section 5.4, the author states two “problems”: “(1) given a transformation, seek a Lagrangian whose functional is invariant; or (2) given a Lagrangian seek transformations that lead to invariance.” For some reason, he calls one or both of them an “inverse problem” which is not the standard terminology used in the calculus of variations, where it refers to the problem of determining whether a given system of differential equations is the Euler-Lagrange equation of a variational principle.^{1} For continuous transformation groups, the solution to both problems was already found by Lie well before Noether appeared on the scene; she was well aware of Lie's contributions. For the second problem, one merely applies a general infinitesimal generator to the Lagrangian to find the infinitesimal determining equations, which can then be solved for the most general symmetry generator. Alternatively one can compute the symmetry group of the associated Euler-Lagrange equations using the standard infinitesimal Lie algorithm, and then determine which ones satisfy the additional variational condition. All of these are straightforward computations, now encoded in computer algebra systems such as Mathematica and Maple. As for the first problem, which is not dealt with here, as Lie proved, the most general invariant Lagrangian is a function of the differential invariants of the group multiplied by an invariant volume element, quantities that Lie (and Noether) were very familiar with, and knew how to find. Indeed, the theory of differential invariants, which is fundamental to the study of invariant variational problems and invariant differential equations,^{1} has never been properly appreciated among the physics community, and the author squandered an opportunity to present it here.

Despite my negative review, there are some aspects of the book I like. As noted previously, the history is quite good, and bringing the career of Emmy Noether to the attention of a broader audience is commendable. The physical exercises and examples are commendable, particularly the material on quantum mechanics. I also like the inclusion of exercises as well as the sections on questions for reflection and discussion—except when they perpetuate some of the author's confusion and inadequate explanations.

But, despite the best of intentions, which I applaud, the bottom line is that Neuenschwander's book does a disservice to both Emmy Noether the mathematician, and her indeed marvelous theorem(s). I am surprised he did not seriously try to address the shortcomings of the first edition, as pointed out for instance in a review by Kosmann-Schwarzbach.^{3} In the preface, he compares Noether's Theorems to a “magnificent summit in an impressive range of ideas,” but unfortunately he has mistaken a lesser peak for the truly majestic mountain towering beyond. She and the physics community deserve much, much better.

## References

*Peter J. Olver is a Professor and Head of the School of Mathematics at the University of Minnesota. He received his Ph.D. in Mathematics from Harvard University in 1976. He is the author of over 140 research papers and 5 books, including undergraduate texts in applied linear algebra and partial differential equations. He was named a “Highly Cited Researcher” by Thomson-ISI in 2003. His research interests revolve around applications of Lie groups and moving frames, and range over image processing, fluid mechanics, quantum mechanics, elasticity, Hamiltonian systems, the calculus of variations, geometric numerical methods, differential geometry, computational algebra, and classical invariant theory*.