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on January 26, 2011
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. Armed with the machinery from the first half of the book and the knowledge of what fields are, the author addresses the question of given gauge invariance what does Noether's theorem tell us about the properties of the Lagrangian.

Like most first addition books, there a few typos in the book but they are easy to spot and do not detract from the beautiful presentation of these subjects.

I hope you enjoy this as much as I did.
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on November 1, 2016
Pedagogic physics starts with specific non-provable rules and derives mathematical statements that lead into mechanics, fields, optics, etc.
These rules include the conservation of energy, momentum, etc. This little gem of a book lays the foundation to understand Emmy Noether's work to try to reverse that understanding to state the Lagrangian or Hamiltonian and then if you can establish symmetry then you can prove those assumed rules. The mathematics is beautiful, but the philosophy is not so firmly established. Instead you may have a tautology where the Hamiltonian is just another statement of the conservation law that you are trying to prove. The foundations of the math itself are not independent of assumption. Frankly, I was delighted to find that I could read and comprehend the level of mathematics presented in this book. For me, the most important point was that conservation laws depend upon symmetry or more simply they have qualifying statements. Thus the evolution of physics is really trying to understand the qualifying statements and the logic that variations imply. Thus, we are not stuck with the "Ten Commandments of Physics" but we may strive to find and establish new and more comprehensive rules about the way things work. By all means, read this book.
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on September 29, 2017
References to the Noether's theorem are aplenty in modern physics books, with various levels of depth depending on a specific usage. This book is special in focusing on the theorem itself and in providing all necessary conceptual environment for complete understanding of its meaning, sources, consequences, and limitations. I enjoyed clear explanations with careful attention to subtle details.
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on December 4, 2011
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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on June 14, 2017
It was recommended to me by a colleague of mine (A. Quadrat), and it was just as he said: "very enlighting". Unlike many physics book that introduce action as kinetic energy minus potential energy without explaining *why*, this book tells you *why*. It nicely complements Landau's courses on theoretical physics (that has the same in-depth perspective), this time with a more mathematical perspective. I appreciated the way the authors asks questions at the beginning of the book and answers them in the different chapters, it completely matches my own way of learning things.
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on April 13, 2012
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. The small morsel provided to me then was something like spatial translation symmetry giving rise to conservation of momentum. At that stage I had no idea how this was proved nor how it fitted into the whole edifice of physics. But it seemed like a big idea. In the 1960s and 1970s you couldn't find a book like Neuenschwander's.

Neuenschwander's book fills in all the gaps as he takes you through the nuts and bolts of the significance of the theorem, its derivation and applications. The maths is at undergraduate level and, despite various typos that others have commented upon, it is a pretty comprehensive treatment for this type of explicatory book. The type of person who buys this book will probably sit down and go through the derivations and verify them (or spot the glitch as the case may be). I am not aware of any other book which goes through this particular subject at so accessible a level. Noether's original paper is harder to digest - if in doubt, try it - there is a translation on the UCLA physics website.

Once you delve into this type of material you begin to wonder too about the fundamental mathematical approximations used to derive the theorems. You begin to appreciate the ubiquity of Taylor's Theorem (applied to produce a local approximation) and wonder that the end results work globally. But that would be another book.

It is interesting to note that a well known PDE expert, Peter Olver, wrote a really negative comment about Neuenschwander's book in a review he wrote of a competing book by Yvette Kosmann-Schwarzbach - “The Noether Theorems”. The review is in the Bulletin of the American Mathematical Society, Vol 50, No 1, January 2013 and it is to be noted that Olver is mentioned for his special help in assisting Yvette Kosmann-Schwarzbach in developing her manuscript. These two books are completely different in their aims. Neuenschwander's book poses all sorts of exercises to develop understanding of what is a deep concept. Criticising this approach is like criticising a formula 1 driver for starting out in GP 3 cars before getting into the more demanding F1 car. Hilariously I have one of Olver's books on PDEs which some PDE experts with a different focus could criticise for being too low beer!

I'd love to see some of the nobodies who have criticised this book write the "definitive" expose so we can judge their talent.
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on December 3, 2012
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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on February 7, 2014
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 covariant derivative, and how particles can be seen to have to satisfy certain requirements on their mass, to mention a few. I've read about these things in expository books like the excellent books by Frank Close. I am mostly self-taught in physics, other than having taken Physics I and II, classical dynamics and General Relativity. The General Relativity course was taken after I had read Steven Weinberg's excellent book on the subject on my own. I know that I have missed many of the small comments that a physics teacher would make during lectures. These small comments can add up to a deeper understanding and so I think I have missed that deeper understanding as well. This book makes me think I am being exposed to what I missed by not learning physics under a teacher at a school.

I could not stop reading this book. I admit that I did not do the problems right away but instead felt compelled to keep reading instead. I figured that I would read the book a few more times and do the problems some time later to really learn it. I even bought two versions of the book. The first version was a paperback and the second was the Kindle book so that I could go back to the book and study it whenever I was near a computer. I have never bought two versions of a book before this.

Before I bought the book, I worried about a couple of reviews, which pointed out some errors. One such review mentioned the incorrect statement of a theorem having to do with the result that the Euler-Lagrange equation describes a function that would minimize an integral. I would swap this minor lapse any day for the clarity of Neuenschwander's presentation. There are many books on the calculus of variations that would state the theorem flawlessly yet would remain devoid of any clarity and motivation. I would classify Neuenschwander's misstatement of the theorem as a typo.

A couple of cool features:

Neuenschwander has a section in which he physically motivates the minus sign in K - U of Hamilton's principle. I bet that this section answers a question that has occurred to everyone dealing with both the Lagrangian and the Hamiltonian. The only other author I remember mentioning anything about the minus sign was John Baez in online student notes of his classical dynamics class. (I wish Neuenschwander would write another book about the physical significance of the Euler-Lagrange equation beyond the standard context of it being used as a tool in variational calculus.)

There's even an appendix which discusses the Legendre transformation and helps one see that there's way more to the correspondence between the Lagrangian and Hamiltonian than a trivial change between variables v and p and a trivial sign change from + to -.

I closely read all the appendices and found them worthwhile.

I like watching great physicists lecture on YouTube. I have really gained a lot by listening to Susskind speak on a whole range of topics. I have watched old lectures by Dirac, which are unbelievably good. I have really enjoyed lectures given by 't Hooft. There are many good expositors on YouTube as well. For instance, I have watched drphysicsA and doctorphys.

I wish that Dwight Neuenschwander would start doing YouTube lectures or maybe teach a course for the Great Courses at the Learning Company. I'm sure that he would have a large and appreciative audience.
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on December 2, 2013
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 for the first time to digest the information in a constructive manner.
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on December 13, 2014
I think the world is just beginning to understand the importance of Emmy Noether's theorem. It might suggest the next logical step after Einstein's theories of relativity, as I discuss in my own work. She is also an excellent role model for women, who may wish to choose mathematics and/or physics as a profession. I first learned about her from Leon Lederman's book, Symmetry and the Beautiful Universe. Lederman is a Nobel prize-winning physicist. Noether was a German mathematician, whose theorem aroudn 1915 was derived from the conservation of energy. While the mathematics is over my head, Lederman sums up the importance of her work in a simple statement: Wherever there is a continuous symmetry, there must be a law of conservation; and wherever there is a law of conservation, there must be a continuous symmetry. In Einstein's special theory of relativity, the principle of the speed of light only applies in a vacuum - it is phenomenological symmetry, not a continuous symmetry. Noether's theorem means that there must be something beyond Einstein's special theory of relativity, and his general theory of relativity doesn't seem to fulfill the bill. Anyone thinking about relativity, should be reviewing Noether's theorem as well; which they can do by purchasing this book. As a mathematician, Noether proved her theorem.
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