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Emmy Noether's Wonderful Theorem [Hardcover]

Dwight E. Neuenschwander
4.0 out of 5 stars  See all reviews (15 customer reviews)

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Book Description

November 16, 2010 0801896932 978-0801896934 1

A beautiful piece of mathematics, Noether's Theorem touches on every aspect of physics. Emmy Noether proved her theorem in 1915 and published it in 1918. This profound concept demonstrates the connection between conservation laws and symmetries. For instance, the theorem shows that a system invariant under translations of time, space, or rotation will obey the laws of conservation of energy, linear momentum, or angular momentum, respectively. This exciting result offers a rich unifying principle for all of physics.

Dwight E. Neuenschwander's introduction to the theorem's genesis, applications, and consequences artfully unpacks its universal importance and unsurpassed elegance. Drawing from over thirty years of teaching the subject, Neuenschwander uses mechanics, optics,geometry, and field theory to point the way to a deep understanding of Noether's Theorem. The three sections provide a step-by-step, simple approach to the less-complex concepts surrounding the theorem, in turn instilling the knowledge and confidence needed to grasp the full wonder it encompasses. Illustrations and worked examples throughout each chapter serve as signposts on the way to this apex of physics.

Noether's Theorem is an essential principle of post-introductory physics. This handy guide includes end-of-chapter questions for review and appendixes detailing key related physics concepts for further study.

Editorial Reviews


Neuenschwander displays the instincts of a good teacher and writes clearly. Using Noether's Theorem as an overarching principle across areas of theoretical physics, he helps students gain a more integrated picture of what sometimes seem to be independent courses—an ever-important thing for undergraduate physics education.

(Dr. Cliff Chancey, University of Northern Iowa)

Neuenschwander writes well and gives thorough explanations.


Without entering into technicalities, the author nevertheless succeeds in preserving a reasonable standard of mathematical rigor and, above all, in convincing the reader of the mathematical beauty and physical relevance of Noether's theorem. If only for that reason, I can strongly recommend this book.

(Frans Cantrijn Mathematical Reviews)

A very readable and concrete introduction to symmetry and invariance in physics with Noether's (first) theorem providing a unifying theme... The style of writing is very engaging and conveys the enthusiasm of the author... The book contains many interesting examples as well as excellent exercises.

(James Vickers London Mathematical Society Newsletter)

About the Author

Dwight E. Neuenschwander is a professor of physics at Southern Nazarene University and editor of the Society of Physics Students Publications of the American Institute of Physics. He won the Excellence in Undergraduate Physics Teaching Award from the American Association of Physics Teachers.

Product Details

  • Hardcover: 264 pages
  • Publisher: Johns Hopkins University Press; 1 edition (November 16, 2010)
  • Language: English
  • ISBN-10: 0801896932
  • ISBN-13: 978-0801896934
  • Product Dimensions: 0.8 x 5.5 x 8.5 inches
  • Shipping Weight: 12.8 ounces (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (15 customer reviews)
  • Amazon Best Sellers Rank: #2,325,580 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
134 of 138 people found the following review helpful
5.0 out of 5 stars A work of Art January 26, 2011
Format:Paperback|Verified Purchase
I am a lifelong student of Physics. I have been a student long beforeI got my PhD in Physics. I am currently a Distinguished Scientist at a Government Lab. This is first review (and possibly the last) I've written for an Amazon book, but I felt compelled to write this after reading this book. It is an excellent example of a 'true' teacher at work who understands how to relate information. This is an art form.

In this book you will learn about Emmy Noether and her work in relating a huge class of conservation laws to nature's symmetries. The book explores how symmetry, invariance and conserved quantities are related, quantitatively. The first half of the book is written for self-study by an undergrad Physics student. It deals predominately with functionals (what are they), functional extremals, and when they are invariant. These chapters are the prelude to Noether's Theorem and Rund-Trautman's version of the theorem. This work first inquires whether a functional is invariant under a given transformation, and if it is, it uses Noether's theorem to get the associated conservation law. Next, it examines the inverse problem; given the transformation can you seek the Lagrangian whose functionals are invariant. In each section the author works examples in some detail and carries these examples with further detail in each of the following chapters. It's like a novel for physicists.

In the last half of the book, the author teaches you how Noether's theorem is used in quantum field theory. He describes the concept of a field through simple examples and introduces Lagrangian densities. Then Noether's theorem is developed for fields and, in particular, quantum fields.
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44 of 47 people found the following review helpful
By Gary
About half-way through, but I already appreciate the biographical information about Noether, and the overview and applications of her results to a variety of problems. The mathematical level is appropriate for upper level undergraduate physics majors and the discussion really helps place the results in context. Nice problems and thought-provoking comments at the end of each chapter. Could use more graphics, and perhaps a little more prose to address the formal mathematical subtleties, but overall, this book admirably fills the largest hole in that small bookshelf containing useful celebrations of deeply significant science and the scientists who created it.
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33 of 35 people found the following review helpful
3.0 out of 5 stars Would like to like it better December 4, 2011
Format:Paperback|Verified Purchase
This book addresses an important gap in the landscape of textbooks on theoretical mechanics. I strongly feel this is the way the subject should be approached as Noether's theorem has such far reaching implications beyond just classical mechanics.

Yet, there are annoying glitches. E.g. the oversight on p.28 with regards to the fundamental lemma of the calculus of variations as has been pointed out in a previous review.

On page 99 the equation (6.3.1) for the Hamiltonian density is incorrect. The way it is written the first term sums over all coordinate indexes. Correct would be to only have time i.e. index zero appear in the first term and sum over all field components if we deal with more than a simple scalar field.

Other times the authors just presents an equation without a modicum of information of how we got there. I.e. the alternative form of the Rund-Trautman identity (RTI II) is given on p.68. It's easy enough to see how the right side follows from RTI I when substituting the canonical variables and using the product rule, but how does the left side of RTI II come about? How does the Euler-Lagrange identity reappear there? (I attached a comment to this review if you are looking for the answer).

Still, I enjoy the book but I would have liked to like it even better.
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6 of 6 people found the following review helpful
5.0 out of 5 stars Very clear. December 3, 2012
Format:Paperback|Verified Purchase
Neuenschwander's superb explanation of Lagrangian and Hamiltonian mechanics will lead you to the best introduction to Noether's theorem I have read.
This well written book is graspable by anyone with multivariate calculus knowledge.
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74 of 101 people found the following review helpful
2.0 out of 5 stars Neuenschwander's Wonderful Lemma July 3, 2011
O.k. help me with this. On p. 28 the author writes (sorry if the math code is hard to read here):

"Lemma (Fundamental Lemma of the Calculus of Variations):
If n(t) is not identically zero in \int_a^b A(t) n(t) dt = 0,
where n(a)=n(b)=0, and
if on the closed interval [a,b] both A(t) and n(t) are twice differentiable,
then A(t) = 0 throughout [a,b]."

This is obviously not true, as seen in the following counterexample:
Let n(t) = sin( 2 pi (t-a)/(b-a) ), and A(t) = 1.
So, n(t) satisfies the stated constraints, and \int_a^b A(t) n(t) dt = \int_a^b sin( 2 pi (t-a)/(b-a) ) dt = 0, yet A(t)=1 is most certainly not 0.

What in the world? Consultation of other author's expositions (e.g. Weinstock's Calculus of Variations or Gelfand and Fomin's Calculus of Variations) reveals that Neuenschwander left out the most crucial phrase in his version of the lemma: "If for EVERY choice of the continuously differentiable function n(t) ..."

How could he be so sloppy in something as fundamental as the Fundamental Lemma of the Calculus of Variations? (I am tempted to make a physicist joke here but I won't).

Once one encounters an error this egregious --- and only on p. 28 --- how can one trust that the author is a faithful guide to this new territory?

And now my mind starts to wonder about other things in the book. I recall the discomfort I felt back on p. 20:

"2.2 Formal Definition of a Functional. Definition: A functional J is a mapping from a set of functions to the real numbers. ...
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Most Recent Customer Reviews
5.0 out of 5 stars Near the top of the list of my favorite books ever
I always wanted to know how physicists arrived at their results when they talk about (local and non-local) gauge theories, advanced ways of looking at Maxwell's equations, the... Read more
Published 5 months ago by readalot
5.0 out of 5 stars Poetically written and catered to the undergraduate.
This is the ultimate pedagogical text for undergraduate level introduction to Noether's Theorem. The author masterfully makes connections that allow students learning the subject... Read more
Published 7 months ago by SirIsaacNewton
1.0 out of 5 stars Not "Mathematics"
This book was basically written for physics students who knows very little mathematics. If you were a beginner physics student who has studied up to Hamiltonian mechanics and is... Read more
Published 15 months ago by butsuritsu
5.0 out of 5 stars A great book about an important subject
The book builds the argument carefully and lays the groundwork for using Noether's Theorem as a test against the validity of any new physical theory. Read more
Published 19 months ago by Kenneth D. Ford
1.0 out of 5 stars bad learning tool
the author never gets the pacing right - his examples veer
between the obvious and the obscure --

in this area there is a dense profusion of notation... Read more
Published 22 months ago by reader
5.0 out of 5 stars Wonderful exposition of Emmy Noether's contributions
I am writing this review primarily to comment on the Kindle version. I do want to say that I am in agreement with other 5-star reviews regarding this book. Read more
Published 22 months ago by Christine
5.0 out of 5 stars Inspiring Exposition
A delightful gem which is a model of physics instruction. The writing is lucid and crisp,
the mathematics is succinctly presented and adroitly complements the elucidated... Read more
Published on May 22, 2012 by G. A. Schoenagel
5.0 out of 5 stars Emmy Noether's Wonderful Theorem
I first heard of Emmy Noether when I was a high school student and how she had proved some abstract theorem which involved symmetry, invariance and conservation of energy. Read more
Published on April 13, 2012 by Peter Haggstrom
5.0 out of 5 stars Nice book to connect the dots
This book did a great job of connecting subtle aspects of classical mechanics, calculus of variations, why tangent spaces are so cherished (also why group theory plays such an... Read more
Published on December 31, 2011 by sankar
4.0 out of 5 stars The importance of the Noether theorem.
The book starts with an analysis about the Eulero-Lagrange theory. The two theories are similar and the back-ground related is the same. Read more
Published on December 12, 2011 by Edoardo Angeloni
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