Noether’s Theorem: Summary and Sources

Dwight E. Neuenschwander, Southern Nazarene University, Bethany OK

These notes offer an overview summary of Noether’s Theorem. Solutions to selected exercises in Emmy Noether’s

Wonderful Theorem (2011) are provided in separate documents.

I. Introduction

Connections between symmetries and conservation laws are among the most elegant and profound relations

in physics. Our formal awareness of them began with the role of space-time symmetries that go back to the

mechanics of Euler and Lagrange. For example, the conservation of energy follows from a system being invariant

under a time translation. In 1918 Amalia “Emmy” Noether (or Nöther) extended such connections to wider domains

of physics. Perhaps because of its original application, physicists who recognize Noether’s theorem by name seem

know it within the context of general relativity or gauge theories of fundamental interactions, propagating the

impression that Noether’s theorem is an exotic point of concern only to specialists. However, Noether’s theorem

can be easily and powerfully applied to topics across the board in the undergraduate physics curriculum, including

classical mechanics, electromagnetism, geometrical optics, special relativity, and non-relativistic quantum

mechanics, in addition to the more advanced topics of gauge theories of particle physics and general relativity.

Noether’s theorem offers a unifying principle for essentially all of physics.

Anyone familiar with the calculus of variations and Lagrangian dynamics is halfway to fluency in

Noether’s theorem. Noether’s theorem holds when a functional is both an extremal and invariant under a

continuous transformation. Section II of these notes reviews the standard notions of variational principles, including

the definition of a functional and the consequences of requiring it to be a maximum or a minimum. Section III

introduces a definition of invariance for a functional, and from it derives Noether’s theorem. Some of the literature

discusses “Noether’s two theorems,” but the distinction between them can be postponed until Section IV, where

Noether’s ideas are applied to fields. Section V briefly relates the context in which the Noether theorem arose and

how it was first used. Section VI contains an annotated set of references for the theorem and biographical sources

about Emmy Noether.

II. Functionals and Extremals

Functionals are mappings that take a function as input and produce a real number as output. This is what

definite integrals do. The functionals encountered in undergraduate physics, such as Fermat’s Principle and

Hamilton’s Principle, are definite integrals of the form

𝐽(𝑞) = ∫ 𝐿[𝑞𝜇(𝑡), �̇�𝜇(𝑡), 𝑡]𝑑𝑡𝑏

𝑎 (1)

where μ = 1,2,…,n, 𝑞�̇� ≡ 𝑑𝑞𝜇

𝑑𝑡 , and L is called the “Lagrangian.” The problem in the calculus of variations is to find

the trajectory—the specific set of generalized coordinates 𝑞𝜇(𝑡)--that make 𝐽 a maximum or a minimum (an

“extremal”). As shown in textbooks on the calculus of variations, the required trajectory is mapped by the 𝑞𝜇 (𝑡)

that satisfy the Euler-Lagrange equation (ELE). The ELE may be written in two equivalent ways,

𝜕𝐿

𝜕𝑞𝜇 − 𝑑

𝑑𝑡

𝜕𝐿

𝜕�̇�𝜇 = 0 (2)

and 𝜕𝐿

𝜕𝑡+

𝑑

𝑑𝑡[�̇�𝜇 𝜕𝐿

𝜕�̇�𝜇 − 𝐿] = 0. (3)

Eq. (3) follows by evaluating dL/dt using the chain rule and invoking Eq. (2). The total time derivatives motivate

the introduction of the canonical momentum and the Hamiltonian. The momentum “conjugate” to qμ is defined as

𝑝𝜇 ≡ 𝜕𝐿/𝜕�̇�𝜇 (4)

which typically is a component of linear momentum if qμ carries a length dimension, and is an angular momentum

component if qμ is an angle. Introduce also the Hamiltonian

𝐻(𝑞𝜇 , 𝑝𝜇 , 𝑡) ≡ �̇�𝜈𝑝𝜈 − 𝐿(𝑞𝜇 , �̇�𝜇 , 𝑡) (5)

(sum repeated indices). The Hamiltonian typically (but not necessarily) turns out to be the system’s energy. In

terms of the momenta and the Hamiltonian, the two version of the ELE become:

𝜕𝐿

𝜕𝑞𝜇 = �̇�𝜇 (6)

and 𝜕𝐿

𝜕𝑡= −�̇�. (7)

By Eq. (6), 𝑝𝜇 is conserved if and only if L contains no explicit dependence on 𝑞𝜇. By Eq. (7), the Hamiltonian will

be conserved if and only if L does not depend explicitly on t. Having come this far, one is now primed for the next

criterion of Noether’s theorem: functional invariance.

III. Functional Invariance

Consider a transformation of the independent variable 𝑡 → 𝑡′ and the dependent variables 𝑞𝜇 → 𝑞′𝜇:

𝑡′ = 𝑇(𝑡, 𝑞𝜈 , ε) (8)

𝑞′𝜇 = 𝑄𝜇(𝑡, 𝑞𝜈 , ε). (9)

The parameter ε could, for example, represent a translation, a boost, or a rescaling; and ε = 0 recovers the identity

transformation. Perform a Taylor series expansion of T and Qμ about ε = 0:

𝑡′ = 𝑡 + 휀𝜏(𝑡, 𝑞𝜈) + 𝑂(휀2)+. .. (10)

𝑞′𝜇 = 𝑞𝜇 + ε휁𝜇 (𝑡, 𝑞𝜈) + 𝑂(ε2)+. .. (11)

where τ and the 휁𝜇 are called the transformation generators. (Some authors use the notation 𝑞′𝜇 = 𝑞𝜇 + 𝛿𝑞𝜇, with

𝛿𝑞𝜇 assumed small, instead of Eq. (11)). The Noether definition of invariance requires that 𝐽′ − 𝐽 can be made as

small as we like by making ε sufficiently small. Formally, a functional is said to be invariant if and only if

𝐽′ − 𝐽 ≡ ∫ 𝐿 (𝑞′𝜇(𝑡′),𝑑𝑞′𝜇

(𝑡′)

𝑑𝑡′ , 𝑡′) 𝑑𝑡′𝑏′

𝑎′− ∫ 𝐿 (𝑞𝜇(𝑡),

𝑑𝑞𝜇(𝑡)

𝑑𝑡, 𝑡) 𝑑𝑡

𝑏

𝑎

~ 휀 𝑠 (12)

as ε → 0, where s > 1. In other words, 𝐽′ − 𝐽 → 0 faster than ε as 휀 → 0. 𝐽′ and 𝐽 can be brought under a common

integral sign and the invariance definition expressed as

𝐿 [𝑞′𝜇(𝑡′),𝑑𝑞′𝜇

(𝑡′)

𝑑𝑡′ , 𝑡′]𝑑𝑡′

𝑑𝑡 − 𝐿 [𝑞𝜇(𝑡),

𝑑𝑞𝜇(𝑡)

𝑑𝑡, 𝑡] ~ εs. (13)

Given a Lagrangian and a transformation, testing for invariance with this definition can be tedious. Around

1970, Hanno Rund and Andrzei Trautman outlined an equivalent but more economical definition: Differentiate Eq.

(13) with respect to ε (recalling Eqs. (10) and (11)) then set ε = 0, resulting in the Rund-Trautman identity (RTI):

(𝜕𝐿

𝜕𝑞𝜇 − 𝑝�̇�) (�̇�𝜇𝜏 − 휁𝜇 ) = 𝑑

𝑑𝑡[𝑝𝜇휁𝜇 − 𝐻𝜏] (14)

(in deriving the RTI the transformed velocities are facilitated by the chain rule, 𝑑𝑞′(𝑡′)

𝑑𝑡′=

𝑑𝑞′

𝑑𝑡

𝑑𝑡

𝑑𝑡′ ≈ (�̇� +

휀휁̇)(1 + 휀�̇�)−1). The RTI serves as both a necessary and a sufficient condition for invariance because the argument

can be reversed. The procedure also generalizes to r parameters, e.g., the Galilean or Poincaré groups, where 𝑡′ =

𝑡 + 휀𝑘𝜏𝑘 + ⋯ and 𝑞′𝜇 = 𝑞𝜇 + 휀𝑘휁𝑘𝜇 + ⋯ with k = 1,2,…,r.

If the functional is extremal and invariant, the ELE and RTI hold simultaneously. Substituting the former

(Eq. 4) into the latter (Eq. 14) immediately yields the Noether conservation law, a superposition of canonical

momenta and the Hamiltonian (or r conservation laws for a set of parameters),

𝑝𝛼휁𝑘𝛼 − 𝐻𝜏𝑘 = 𝑐𝑜𝑛𝑠𝑡. (15)

Noether’s theorem includes not only the conservation laws for energy, linear momentum, and angular

familiar from Lagrangian mechanics, but also reveals conservation laws for systems that, at first glance, appear to

have none. For instance, in the damped oscillator neither momentum nor mechanical energy are conserved. The

Lagrangian 𝐿(𝑥, �̇�, 𝑡) = [1

2𝑚�̇�2 −

1

2𝑘𝑥2] 𝑒𝑏𝑡/𝑚 in the ELE produces the equation of motion, 𝑚�̈� + 𝑏�̇� + 𝑘𝑥 = 0. A

time translation t’ = t + ε and spatial rescaling 𝑥 ′ = 𝑥 (1 − 휀𝑏

2𝑚) satisfies the RTI, leading to [

1

2𝑚�̇�2 +

1

2𝑘𝑥2 +

1

2𝑏𝑥�̇�] 𝑒

𝑏𝑡

𝑚 = 𝑐𝑜𝑛𝑠𝑡.

The RTI can also be used to determine a family of Lagrangians whose functionals are invariant under a

given transformation. Alternatively, given a Lagrangian, transformations that lead to conservation laws can be

systematically found by imposing the RTI and solving for the generators. In that case one writes the RTI as a

polynomial in powers of velocity (using the chain rule on �̇� and 휁̇). Since the RTI must hold whatever the velocity,

the coefficients of the various powers of velocity are set to zero, producing the so-called Killing equations that are

solved for the generators.

A more liberal definition of invariance allows a term linear in ε,

𝐿′ 𝑑𝑡′

𝑑𝑡− 𝐿 = 휀

𝑑𝜙

𝑑𝑡 + O(휀𝑠) (16)

with s > 1, leading to an extended Noether conservation law,

𝑝𝛼휁𝛼 − 𝐻𝜏 − 𝜙 = 𝑐𝑜𝑛𝑠𝑡. (17)

An elementary example includes projectile motion in the xy plane without air resistance, having the Lagrangian

𝐿(𝑦, �̇�, �̇�) = 1

2𝑚(�̇�2 + �̇�2) − 𝑚𝑔𝑦. Consider the simplest Galilean transformation, 𝑡′ = 𝑡, 𝑥 ′ = 𝑥 − 𝑣𝑡, 𝑦′ = 𝑦,

where the relative velocity v between two inertial frames assumes the role of ε. We find 𝐿′ 𝑑𝑡′

𝑑𝑡− 𝐿 = −𝑚�̇�𝑣 +

𝑂(𝑣2), an instance of Eq. (16) with �̇� = −𝑚�̇�. The resulting conservation law expresses conservation of the x-

component of momentum. Such a liberalized definition of invariance is called “divergence invariance” because

𝜕𝜈𝜙𝑘𝜈 appears in the Noether conservation law arising from multiple-integral functionals, which we examine next.

IV. Multiple-Integral Functionals

To illustrate the emergence of a multiple-integral functional, treat a harmonic wave on a string as a row of

simple harmonic oscillators, where each bit of string has infinitesimal length dx and Lagrangian ℒ. The Lagrangian

for the entire string of length l is 𝐿 = ∫ ℒ 𝑑𝑥𝑙

0. Therefore J = ∫ 𝐿𝑑𝑡

𝑏

𝑎= ∫ ∫ ℒ

𝑏

𝑎

𝑙

0 𝑑𝑥𝑑𝑡. More generally, consider n

independent variables 𝑞𝜈and m dependent variables 𝐴𝜇 = 𝐴𝜇(𝑞𝜈) in a functional integrated over a domain ℛ,

𝐽 = ∫ ℒ 𝑑𝑛𝑞ℛ

(18)

where ℒ is called the Lagrangian density, a function of field components 𝐴𝜇(𝑞𝜈), their derivatives 𝜕𝜈𝐴𝜇 ≡ 𝐴 ,𝜈𝜇

and

the 𝑞𝜈. The ELE generalizes from Eq. (4) into

𝜕ℒ

𝜕𝐴𝜇 = 𝜕𝜈(𝜕ℒ

𝜕𝐴 ,𝜈𝜇 ) (19)

which suggests the introduction of the field theory version of canonical momentum,

𝑝 𝜈𝜇

≡ 𝜕ℒ

𝜕𝐴 ,𝜇𝜈 . (20)

The ELE in terms of a Hamiltonian density generalizes Eq. (3) to

𝜕𝜇ℒ = −𝜕𝜈ℋ 𝜇𝜈 (21)

with Hamiltonian tensor ℋ 𝜇𝜈 = 𝐴 ,𝜆

𝜈 𝑝 𝜇𝜆 − 𝛿 𝜇

𝜈 ℒ. Under an r-parameter family of transformations

𝑞′𝜇 = 𝑞𝜇 + 휀𝑘 𝜏𝑘𝜇

+ ⋯ (22)

𝐴′𝜇 = 𝐴𝜇 + 휀𝑘 휁𝑘𝜇 + ⋯ (23)

and allowing for divergence-invariance, the RTI becomes, for each k,

[ 𝜕ℒ

𝜕𝐴𝜇 − 𝜕𝜌 (𝜕ℒ

𝜕𝐴 ,𝜌𝜇 )] (𝐴 ,𝜈

𝜇 𝜏𝑘𝜈 − 휁𝑘

𝜇 ) = 𝜕𝜈[ 𝑝 𝜌𝜈 휁𝑘

𝜌 − ℋ 𝜌𝜈 𝜏𝑘

𝜌 − 𝜙𝑘𝜈 ]. (24)

When the ELE (Eq. 19) also holds, the Noether conservation law becomes an equation of continuity,

𝜕𝜈[𝑝 𝜆𝜈 휁𝑘

𝜆 − ℋ 𝜆𝜈 𝜏𝑘

𝜆 − 𝜙𝑘𝜈] = 0. (25)

In addition to energy-momentum conservation from spacetime transformations, important applications also include

gauge transformations, where 𝑞′𝜇 = 𝑞𝜇 but the fields are phase-shifted,

𝐴′𝜇 = 𝑒 𝑖𝑔𝑘𝜀𝑘𝐴𝜇 = 𝐴𝜇 + 𝑖𝑔𝑘휀𝑘𝐴𝜇 + O(휀2), (26)

where 𝑔𝑘 denotes a coupling constant such as electric charge. In a global gauge transformation the εk are

independent of spacetime coordinates, and the Noether equation of continuity asserts local charge conservation (or

Ward identities in quantum theory versions).

Noether’s “second theorem” allows the 휀𝑘 to be functions of the independent variables. If 휀𝑘 = 휀𝑘(𝑞𝜈),

the field phase changes by different amounts at different events in spacetime—a local gauge transformation. In that

case the 𝜕𝜈𝐴𝜇 terms in the Lagrangian density spoil local gauge invariance, because ℒ′ picks up (𝜕𝜈휀𝑘)𝐴𝜇 terms that

did not appear in ℒ. Instead of a vanishing divergence, the terms that appear in the corresponding version of Eq.

(25) yield a set of constraints (e.g., the Bianchi identities in general relativity). The remedy requires the derivative to

be modified into a “covariant derivative” by adding a compensating term to the usual derivative. Then Noether’s

second theorem gives a conservation law expressed as a vanishing covariant divergence. In gauge theories of

particle physics the compensating terms describe matter fields coupling to spin-1 gauge bosons.

Since Noether’s “second theorem” is the version known to experts in quantum field theory and general

relativity, the false impression may have grown over the years that these topics form the extent of Noether theorem

applications to physics. That Noether’s theorem also applies to mechanical oscillators, electric circuits, geometrical

optics, nonrelativistic quantum mechanics, classical electrodynamics, and geometry, seems to have escaped the

notice of most physicists. (Although the Feynman Lectures discusses the calculus of variations, Feynman was silent

on Noether’s theorem.) Because Noether’s theorem (especially her “first theorem” ) is so accessible to

undergraduate physicists—who have already been introduced to Lagrangian mechanics—Noether’s theorem can and

should be more widely known among all physicists. It offers a unifying principle that cuts across all the physics

disciplines.

V. The Genesis of Noether’s Theorem

Albert Einstein introduced general relativity in 1915, and almost immediately David Hilbert derived

Einstein’s field equations from a variational principle. However, Hilbert encountered a puzzle regarding energy

conservation, which classical mechanics connected to invariance under time translations. But in general relativity a

global time coordinate does not exist. Hilbert asked Emmy Noether to investigate.

At Göttingen she had recently finished her PhD in abstract algebra, on the subject of bilinear invariant

theory, with Paul Gordon as her advisor. She had learned mathematics from the great mathematicians of the day,

including Gordon, Hilbert, Herman Minkowski, and Felix Klein. She was recognized as a mathematician of

considerable brilliance. Hilbert had argued on her behalf against the cultural reluctance of that time against the

hiring of women professors.

Noether’s response to Hilbert’s inquiry was published in 1918. She applied transformation group theory to

general relativity and created a formalism whose capabilities went far beyond resolving Hilbert’s problem.

Noether’s remedy to Hilbert’s energy question was to make time transformations local, which set the stage for gauge

theories. This distinction between global and local transformations led Noether and subsequent authors to speak of

the “two theorems.”

In a letter December 27, 1918, Albert Einstein wrote to Felix Klein, “”On receiving the new work from

Fräulein Nöther, I again find it a great injustice that she cannot lecture officially. I would be very much in favor of

taking energetic steps in the ministry [to overturn this rule].” In 1919 the university relented and granted Noether an

untenured professorship, and began paying her a modest salary in 1923. When the Nazis came to power in 1933,

Noether was among the first Jewish intellectuals to be dismissed. She accepted a position at Bryn Mawr College in

Pennsylvania, and lectured at the new Institute for Advanced Study in Princeton, New Jersey. Unfortunately, in

1935, following a successful operation to remove a tumor, Emmy Noether evidently contracted a virulent infection

and passed away on April 14, 1935, at the age of 57. Einstein wrote in a New York Times obituary that Emmy

Noether was “the most significant creative mathematical genius thus far produced since the higher education of

women began.” Her ashes were buried in the cloister of the Bryn Mawr library. In 1980 the Association of Women

in Mathematics initiated the annual Emmy Noether Lecture, established “to honor women who have made

fundamental and sustained contributions to the mathematical science.”

VI. Selected References

For a more comprehensive set of reference (~ 70) consult the AJP Physics Resource Letter, “Noether’s Theorem in

the Undergraduate Curriculum” by DEN (in press, American Journal of Physics).

Seminal papers

1. “Invariante variationsprobleme,” E.A. Noether, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys.

Klasse II, 235-257 (1918). Emmy Noether’s celebrated 1918 paper. For a translation see “Invariant variation

problems,” E.A. Noether and M. Tavel (tr.), Transp. Theory Stat. Phys. 1, 186-207 (1971).

2. “Über die Erhaltungssätze der Elecktrodynamik,” E. Bessel-Hagen, Math. Ann. 84, 258-276 (1921).

Conservation laws in electrodynamics from conformal invariance, introducing divergence invariance from “an oral

communication by Fraülein Emmy Nöther herself.”

3. “Hamilton’s Principle and the Conservation Theorems of Mathematical Physics,” E.L. Hill, Rev. Mod. Phys. 23

(3), 253 -260 (1951). An oft-cited exposition on Noether’s theorem.

4. “A direct approach to Noether’s theorem in the calculus of variations,” H. Rund, Util Math. 2, 205-214 (1972).

5. “Noether’s equations and conservation laws,” A. Trautman, Comm. Math. Phys. 6, 248-261 (1967).

Anthologies

6. Emmy Noether: Gesammelte Abhandlungen (Collected Papers), edited by H. von N. Jacobson (Springer-

Verlag, Berlin, 1983). Noether’s mathematical papers, in German. Includes text, in English, of a moving address

(pp. 1-11), “In Memory of Emmy Noether,” delivered by P.S. Alexandrov on 5 September 1935.

7. Emmy Noether in Bryn Mawr: Proceedings of a Symposium sponsored by the Association for Women in

Mathematics in honor of Emmy Noether’s 100th Birthday, B. Srinivasan and J.D. Sally, eds. (Springer-Verlag,

New York, 1983). Paper 6 by K. Uhlenbeck (pp. 103-115) describes the “first” Noether’s theorem. Includes

biographical notes and personal recollections, “Emmy Noether at Bryn Mawr,” pp. 139-146; G.C. Quinn et.al., and

“Emmy Noether: Historical Contexts”, U.C. Merzbach, pp. 161-171.

8. The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century, Y. Kosmann-

Schwarzbach, translation by B.E. Schwarzbach. (Springer, New York, 2011). A very thorough study of the impact

of Noether’s theorem across time, including chapters on the theorem’s inception, its reception by Noether’s

contemporaries and by historians: “The following testimony confirms the lack of a precise and direct knowledge of

Noether’s work among the best physicists as late as 1960...” (p. 85), eras in the theorem’s appreciation, such as Ch.

4, “…from Bessel-Hagen to Hill,” its reception since 1950 (Hill), and “genuine generalizations” after 1970

(Rund/Trautman).

Tutorials

9. “E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws,” N. Byers, in

Israel Mathematical Conference Proceedings 12 (1999), Symposium on the Heritage of Emmy Noether in Algebra,

Geometry, and Physics, Bar Ilan University, Tel Aviv, Israel, 2-3 December 1996.

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html and arXiv:physics/9807044v2, 23 September

1998. Historical background to, and a qualitative synopsis of, Noether’s 1918 paper. I recommend reading this

paper prior to or concurrently with Ref. 1.

10. “Symmetries, Conservation Laws, and Noether’s Theorem,” D.E. Neuenschwander, SPS Newsletter (January

1996), 14-16 and Radiations, (Fall 1998), 12-15. Bare-bones introduction, written for undergraduates.

11. “Symmetries and conservation laws: Consequences of Noether’s theorem,” J. Hanc, S. Tuleja, and M. Hancova,

Am. J. Phys. 72 (4), 428-435 (2004). Bibcode:2004AmJPh..72..428H. doi:10.1119/1.1591764. Written for

introductory physics students, breaks a particle trajectory into straight-line segments, includes computer graphics.

Textbooks on Noether’s Theorem

12. Invariant Variational Principles, J.D. Logan (Academic Press, New York, 1977). A textbook on Noether’s

theorem adapted from lectures delivered in special topics courses in applied mathematics, intended for upper-

division undergraduate or graduate students in mathematics, physics, and applied sciences.

13. Emmy Noether’s Wonderful Theorem, D.E. Neuenschwander (Johns Hopkins University Press, Baltimore,

2011). An undergraduate textbook on Noether’s theorem for physics majors, adapted from use in undergraduate

mechanics courses, with extensions to electrodynamics and other field theories with gauge invariance.

Textbooks with a chapter or section on Noether’s Theorem.

14. Methods of Mathematical Physics, Vol. 1, R. Courant and D. Hilbert (Springer, Berlin, 1937; Interscience

Publishers, New York, 1953), pp. 262-266.

15. Electrodynamics and the Classical Theory of Fields and Particles, A.O. Barut (Macmillan, New York,

1964), pp. 99-131.

16. Classical Mechanics, H. Goldstein (2nd ed., Addison-Wesley, Reading, MA, 1980), pp. 588-596.

17. Applied Mathematics: A Contemporary Approach, J.D. Logan (Wiley-Interscience, New York,

1987), Ch. 7, “Noether’s Theorem,” pp. 416-488. Very accessible, sadly not carried over in later editions.

18. Quantum Field Theory, M. Srednicki (Cambridge University Press, Cambridge, 2007). Ch. 21, “The quantum

action,” pp. 127-131; Ch. 22, “Continuous symmetries and conserved currents,” pp. 132-139.

Journal Articles on Applications of Noether’s Theorem

Arranged in chronological order, most include summaries of Noether’s Theorem.

19. “Derivation of conserved quantities from symmetries of the Lagrangian in field theory,” T. Boyer, Am. J. Phys.

34 (6) 475- 479 (1966).

20. “Invariant Theory of Variational Problems for Geometrical Optics,” H. Rund, Tensor 18, 239-258 (1967).

21. “Conservation laws for gauge-invariant Lagrangians in classical mechanics,” J. Lèvy-Leblond, Am. J. Phys. 39

(5), 502-506 (1971). Applies divergence-invariance to classical mechanics.

22. “Noether’s Theorem in classical mechanics,” J. Ray, Am.J.Phys. 40 (3),493-494(1971). Comments on Ref. 21.

23. “Invariance and the n-Body Problem,” J.D. Logan, J. Math. Anal. App. 43 (1), 191-197 (1973). Imposes

invariance identities to constrain Lagrangians exhibiting invariance under the Galilean group of transformations.

24. “On Some Invariance Identities of H. Rund,” J.D. Logan, Util. Math. 7, 281-286 (1975). Justifies the strategy

of inverting the RTI to find the generators; illustrated with Emden’s equation.

25. “Noether’s theorem in classical mechanics,” E.A. Desloge and R.I. Karch, Am. J. Phys. 45 (4), 336-340 (1977).

Succinct but busy notation; includes divergence-invariance and Galilean transformations.

26. “Conservation Laws in Circuit Theory,” J.D. Logan, Int. J. Elect. Enging. Educ., 17, 349-354 (1980). Imposes

the RTI to find generators and conservation law for the series RLC circuit.

27. “The Application of Noether's Theorem to Optical Systems,” J.W. Blaker and M.A. Tavel, Am. J. Phys. 42,

857-861 (1974).

28. “Noether’s theorem in discrete classical mechanics,” N. Bobillo-Ares, Am. J. Phys. 56 (2), 174-177 (1989).

Noether’s theorem developed for systems of particles using methods similar to those for fields.

29. “A Hamiltonian for geometrical optics,” B. Turner, J. Undergrad. Res. Phys. 10, 23-28 (1991). Includes a

Noether theorem application as an example.

30. “Adiabatic invariance derived from the Rund-Trautman Identity and Noether's Theorem,” S. Starkey and D.E.

Neuenschwander, Am. J. Phys. 61 (11), 1008-1013 (1993). Find generators that lead to adiabatic invariance in

classical mechanics applications.

31. “The adiabatic invariants of plasma physics derived from the Rund-Trautman Identity and Noether's Theorem,”

G. Taylor and D.E. Neuenschwander, Am. J. Phys. 64 (11), 1428- 1430 (1996).

32. “Comment on ‘The adiabatic invariants of plasma physics derived from the Rund-Trautman identity and

Noether’s theorem,’” G. Taylor, D.E Neuenschwander, and C. Ferrario, Am. J. Phys. 66 (11), 1016-1017 (1998).

33. “Noether’s theorem, rotating potentials, and Jacobi’s integral of motion,” C.M. Giordano and A.R. Plastino, Am.

J. Phys. 66 (11), 989-995 (1998). Derives Jacobi’s integral from Noether’s theorem.

34. “Homage to Emmy Noether,” D. Haylebrouck, I, Hargitti, M. Hargittai, Math. Intell. 21 (1), 48-49 (2002).

35. “Noether’s Theorem in a Rotating Reference Frame,” L. Dallen and D.E. Neuenschwander, Am. J. Phys. 79 (3),

326-332 (2011). Generators and Noether conservation laws found for mechanics in rotating frame; contains results

of Ref. 33 as special case.

Biography

36. Hilbert, C. Reid (Springer-Verlag, New York, 1972). Biography of David Hilbert, but Emmy Noether makes

several appearances. “[S]he had an impressive knowledge of certain subjects which Hilbert and Klein needed for

their work on relativity theory, and they were both determined that she must stay in Göttingen.” (p. 143).

37. Women in Mathematics, L.M. Osen (MIT Press, Cambridge, MA, 1974). An ensemble of biographical essays.

Noether (pp. 141-152) keeps distinguished company with Emilie de Breteuil, Caroline Herschel, and five others.

38. The Association of Women in Mathematics sponsors the annual Emmy Noether Lecture, founded in 1980 “to

honor women who have made fundamental and sustained contributions to the mathematical sciences.” See

https://sites.google.com/site/awmmath/programs/noether-lectures

39. Emmy Noether: A Tribute to Her Life and Work, J.W. Brewer and M.K. Smith, eds. (Marcel Dekker, Inc.,

New York and Basel, 1981). Eleven writers contributed six biographical chapters and five mathematical chapters.

Chapter 7, “The Calculus of Variations,” pp. 125-130, was written by E.J. McShane, who knew Noether personally:

“…Perhaps Emmy Noether’s attention was drawn to such conservation laws by her physicist brother. It did not

occur to me to ask her about this when I used to meet her, in the early 1930s; in fact, if it had occurred to me, I

probably would have lacked the presumption of asking such a question of a mathematician so much older and more

distinguished than I, even though that mathematician was the friendly and approachable Emmy Noether.”

40. Emmy Noether: 1882-1935, Auguste Dick (Birkhäuser Verlag, Basel, German edition 1970, English

translation by H.I. Blocher, 1981). A concise but detailed biography of Noether, with photos, obituaries, lists of her

publications and doctoral students.

41. ‘Subtle is the Lord…’ The Science and the Life of Albert Einstein, A. Pais (Oxford University Press, New

York, 1982). Ch. 14, pp. 258-259 and 274-276, describes the interactions between Einstein and Hilbert in 1915

with the advent of general relativity, and Noether’s role in clarifying local energy conservation. Albert Einstein

held Emmy Noether in great esteem.

42. Remarkable Mathematicians, I. James (Cambridge University Press, Cambridge, 2002). Noether appears on

pp. 321-326. “We now come to the first woman mathematician who can undoubtedly be described as ‘great.’…So

what was she like personally? ‘Warm like a loaf of bread,’ Weyl wrote…” (p. 325). (E)

43. “This Month in Physics History: March 23, 1882: Birth of Emmy Noether,” APS News 22 (3), 2 (2013).

Executive summary of Noether’s life and work.

Books (or sections of books) on the Calculus of Variations:

44. The Feynman Lectures on Physics, R.P. Feynman, R.B. Leighton, and M. Sands (Addison-Wesley, Reading,

MA, 1963), Vol. II, Ch. 19, “The Principle of Least Action,” pp. 19-1 – 19-14.

45. Calculus of Variations, I.M. Gelfand and S.V. Fomin; translated and edited by R.A. Silverman, (Dover

Publications, Mineola, NY, 1963, 1991). Noether’s theorem for single dependent variable is presented on pp. 79-83;

multiple dependent variables on pp. 176-179, with subsequent applications to field theory.

46. Mathematical Methods in the Physical Sciences, M.L. Boas (John Wiley & Sons, New York, 1966). Ch. 8,

“Calculus of Variations” pp. 368-396.

47. Perfect Form: Variational Principles, Methods, and Applications in Elementary Physics, D.S. Lemons

(Princeton University Press, Princeton, 1997). A diverse collection of worked examples.

48. Any textbook on classical mechanics will include a section or chapter on the calculus of variations and

Lagrangian-Hamiltonian dynamics. E.g., see Classical Dynamics of Particles and Systems, J.B. Marion and S.T.

Thornton (5th ed., Brooks/Cole, Belmont, CA, 2004), Chs. 6-7; Classical Mechanics, J.R. Taylor (University

Science Books, Sausalito, CA, 2005), Chs. 6-7.