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Emmy Noether's Wonderful Theorem


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Table of Contents

PrefaceAcknowledgmentsQuestionsI. WHEN FUNCTIONALS ARE EXTREMAL1. Symmetry1.1. Symmetry, Invariances, and Conservation Laws1.2. Meet Emmy Noether2. Functionals2.1. Single-Integral Functionals2.2. Formal Definition of a Functional3. Extremals3.1. The Euler-Lagrange Equation3.2. Conservation Laws as Corollariesto the Euler-Lagrange Equation3.3. On the Equivalence of Hamilton's Principleand Newton's Second Law3.4. Where Do Functional Extremal PrinciplesCome From?3.5. Why Kinetic Minus Potential Energy?3.6. Extremals with External ConstraintsII. WHEN FUNCTIONALS ARE INVARIANT4. Invariance4.1. Formal Definition of Invariance4.2. The Invariance Identity4.3. A More Liberal Definition of Invariance5. Emmy Noether's Elegant (First) Theorem5.1. Invariance + Extremal = Noether's Theorem5.2. Executive Summary of Noether's Theorem5.3. "Extremal" or "Stationary"?5.4. An Inverse Problem5.5. Adiabatic Invariance in Noether's TheoremIII. THE INVARIANCE OF FIELDS6. Noether's Theorem and Fields6.1. Multiple-Integral Functionals6.2. Euler-Lagrange Equations for Fields6.3. Canonical Momentum and the HamiltonianTensor for Fields6.4. Equations of Continuity6.5. The Invariance Identity for Fields6.6. Noether's Theorem for Fields6.7. Complex Fields6.8. Global Gauge Transformations7. Local Gauge Transformations of Fields7.1. Local Gauge Invariance and Minimal Coupling7.2. Electrodynamics as a Gauge Theory,Part 17.3. Pure Electrodynamics, Spacetime Invariances,and Conservation Laws7.4. Electrodynamics as a Gauge Theory,Part 27.5. Local Gauge Invariance and Noether Currents7.6. Internal Degrees of Freedom7.7. Noether's Theorem and GaugedInternal Symmetries8. Emmy Noether's Elegant (Second) Theorem8.1. Two Noether Theorems8.2. Noether's Second Theorem8.3. Parametric Invariance8.4. Free Fall in a Gravitational Field8.5. The Gravitational Field Equations8.6. The Functionals of General Relativity8.7. Gauge Transformations on Spacetime8.8. Noether's Resolution of an Enigma inGeneral RelativityIV. TRANS-NOETHER INVARIANCE9. Invariance in Phase Space9.1. Phase Space9.2. Hamilton's Principle in Phase Space9.3. Noether's Theorem and Hamilton's Equations9.4. Hamilton-Jacobi Theory10. The Action as a Generator10.1. Conservation of Probabilityand Continuous Transformations10.2. The Poetry of NatureAppendixesA. Scalars, Vectors, and TensorsB. Special RelativityC. Equations of Motion in Quantum MechanicsD. Conjugate Variables and Legendre TransformationsE. The JacobianF. The Covariant DerivativeBibliographyIndex

About the Author

Dwight E. Neuenschwander is a professor of physics at Southern Nazarene University. He is a columnist for the Observer, the magazine of the Society for Physics Students, and the author of Tensor Calculus for Physics: A Concise Guide.


As this book is well written and contains a very good set of exercises, it can serve as the primary text for a special topics course. * Choice *
Nadis gives no technical details, but Neuenschwander does, in a book for physics majors with a strong background in mathematics; the book does not shy away from Lie groups and the study of invariants. This new edition delves into distinctions between two Noether theorems and adds more exercises, references, and details. * Mathematics Magazine *
Neuenschwander sets out from the beginning to help the reader who must be familiar with calculus and a few other standard topics, but who is not yet fluent in these areas... His role is to be the teacher on the side, prompting the reader with interesting observations and questions... He anticipates problems, guides you yet also makes you think things through... Not only a very worthwhile read for its content but also for its style. -- Ken Zetie, St. Paul's School * Mathematical Gazette *
Well-written... Throughout there is reference to the life of Emmy Noether, including the many difficulties related to being a woman in a man's world... I am glad her story is given an airing here as she fails to be as famous as she undoubtedly should be. -- Phil Dyke, FIMA * Mathematics Today *
Technical and yet ultimately poetic book on Emmy Neother's wonderful theorems... Neuenschwander's work is recommended for anyone who wants to gain a deeper understanding and appreciation of the physics and mathematics behind Emmy Noether's work, as well as the particular challenges she faced in her life. -- Miriam R. Aczel, Centre for Environmental Policy, Imperial College London * Contemporary Physics *

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