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Emmy Noether's Wonderful Theorem [NOOK Book]

Overview

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 ...

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Emmy Noether's Wonderful Theorem

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Overview

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.

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Editorial Reviews

Mathematical Reviews - Frans Cantrijn
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.
London Mathematical Society Newsletter - James Vickers
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.
Choice
Neuenschwander writes well and gives thorough explanations.
Choice

Neuenschwander writes well and gives thorough explanations.

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Product Details

  • ISBN-13: 9780801899454
  • Publisher: Johns Hopkins University Press
  • Publication date: 12/29/2010
  • Sold by: Barnes & Noble
  • Format: eBook
  • Pages: 264
  • Sales rank: 1,129,031
  • File size: 24 MB
  • Note: This product may take a few minutes to download.

Meet 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.

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Table of Contents

Preface xi

Acknowledgments xiii

Glossary of Symbols xv

Primary and Auxiliary Questions xvii

1 Prologue 1

1.1 Symmetry, Invariance, and Conservation Laws 1

1.2 Emmy Noether Biographical Notes 6

Part I When Functionals Are Extremal

2 Functionals 15

2.1 Single-Integral Functionals 15

2.2 Formal Definition of a Functional 20

3 Extremals 25

3.1 The Euler-Lagrange Equation 25

3.2 Corollaries to the Euler-Lagrange Equation 31

3.3 On the Equivalence of Hamilton's Principle and Newton's Second Law 36

3.4 Where Did Hamilton's Principle Come From? 39

3.5 Why Kinetic Minus Potential Energy? 47

3.6 Extremals with External Constraints 49

Part II When Functionals Are Invariant

4 Invariance 61

4.1 Formal Definition of Invariance 61

4.2 Condition for Invariance: The Rund-Trautman Identity 66

4.3 A More Liberal Definition of Invariance 68

5 Emmy Noether's Elegant Theorem 72

5.1 Extremal + Invariance = Noether's Theorem 72

5.2 The Inverse Problem: Finding Invariances 75

5.3 Adiabatic Invariance and Noether's Theorem 79

Part III The Invariance of Fields

6 Fields and Noether's Theorem 91

6.1 Multiple-Integral Functionals 91

6.2 Euler-Lagrange Equations for Fields 96

6.3 Canonical Momenta and the Hamiltonian for Fields 99

6.4 Equations of Continuity 101

6.5 The Rund-Trautman Identity for Fields 103

6.6 Noether's Theorem for Fields 107

6.7 Complex Fields 108

6.8 Global Gauge Transformations 113

7 Gauge Invariance as a Dynamical Principle 125

7.1 Local Gauge Invariance and the Covariant Derivative 125

7.2 Electrodynamics as a Gauge Theory I: Field Tensors 129

7.3 Pure Electrodynamics, Spacetime Invariances, and Conservation Laws 135

7.4 Electrodynamics as a Gauge Theory II: Sources and Minimal Coupling 140

7.5 Internal Degrees of Freedom 143

7.6 Non-Abelian Gauge Transformations 153

Part IV Post-Noether Invariance

8 Invariance in Phase Space 171

8.1 Phase Space 171

8.2 Hamilton's Principle in Phase Space 172

8.3 Noether's Theorem through Hamilton's Equations 175

8.4 Hamilton-Jacobi Theory 177

9 The Action as a Generator 191

9.1 Conservation of Probability and Unitary Transformations 192

9.2 Continuous Spacetime Transformations in Quantum Mechanics 195

9.3 Epilogue 200

Appendixes

A Scalars, Vectors, Tensors, and Coordinate Transformations 205

B Special Relativity 211

C Equations of Motion in Quantum Mechanics 217

D Legendre Transformations and Conjugate Variables 221

E The Jacobian 225

Bibliography 229

Index 235

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