Emmy Noether's wonderful theorem

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 more than 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. Emmy Noether’s Wonderful Theorem......Page i frontispiece: Emmy Noether......Page ii ISBN : 978-0-8018-9693-4......Page iv C0NTENTS......Page vii Preface......Page xi Acknowledgments......Page xiii Glossary of Symbols......Page xv Primary and Auxiliary Questions......Page xvii 1. Prologue......Page 1.djvu 1.1. Symmetry, invariance, and Conservation Laws......Page 1.djvu 1.2. Emmy Noether Biographical Notes......Page 6.djvu I WHEN FUNCTIONALS ARE EXTREMAL......Page 13.djvu 2. Functionals......Page 15.djvu 2.1. Single-Integral Functionals......Page 15.djvu 2.2. Formal Definition of a Functional......Page 20.djvu 3. Extremals......Page 25.djvu 3.1. The Euler-Lagrange Equation......Page 25.djvu 3.2. Corollaries to the Euler-Lagrange Equation......Page 31.djvu 3.3. On the Equivalence of Hamilton’s Principle and Newton’s Second Law......Page 36.djvu 3.4. Where Did Hamilton’s Principle Come From?......Page 39.djvu 3.5. Why Kinetic Minus Potential Energy?......Page 47.djvu 3.6. Extremals with External Constraints......Page 49.djvu II WHEN FUNCTIONALS ARE INVARIANT......Page 59.djvu 4. Invariance......Page 61.djvu 4.1. Formal Definition of Invariance......Page 61.djvu 4.2. Condition for Invariance: The Rund-Trautman Identity......Page 66.djvu 4.3. A More Liberal Definition of Invariance......Page 68.djvu 5. Emmy Noether’s Elegant Theorem......Page 72.djvu 5.1. Extremal + Invariance = Noether’s Theorem......Page 72.djvu 5.2. The Inverse Problem: Finding lnvariances......Page 75.djvu 5.3. Adiabatic invariance and Noether's Theorem......Page 79.djvu III THE INVARIANCE OF FIELDS......Page 89.djvu 6. Fields and Noether’s Theorem......Page 91.djvu 6.1. Multiple—lntegral Functionals......Page 91.djvu 6.2. Euler-Lagrange Equations for Fields......Page 96.djvu 6.3. Canonical Momenta and the Hamiltonian for Fields......Page 99.djvu 6.4. Equations of Continuity......Page 101.djvu 6.5. The Rund—Trautman Identity for Fields......Page 103.djvu 6.6. Noether's Theorem for Fields......Page 107.djvu 6.7. Complex Fields......Page 108.djvu 6.8. Global Gauge Transformations......Page 113.djvu 7. Gauge Invariance as a Dynamical Principle......Page 125.djvu 7.1. Local Gauge lnvariance and the Covariant Derivative......Page 125.djvu 7.2. Electrodynamics as a Gauge Theory I: Field Tensors......Page 129.djvu 7.3. Pure Electrodynamics, Spacetime lnvariances, and Conservation Laws......Page 135.djvu 7.4. Electrodynamics as a Gauge Theory II: Sources and Minimal Coupling......Page 140.djvu 7.5. Internal Degrees of Freedom......Page 143.djvu 7.6. Non-Abelian Gauge Transformations......Page 153.djvu IV POST-NOETHER INVARIANCE......Page 169.djvu 8. Invariance in Phase Space......Page 171.djvu 8.1. Phase Space......Page 171.djvu 8.2. Hamilton's Principle in Phase Space......Page 172.djvu 8.3. Noether's Theorem through Hamilton's Equations......Page 175.djvu 8.4. Hamilton—Jacobi Theory......Page 177.djvu 9. The Action as a Generator......Page 191.djvu 9.1. Conservation of Probability and Unitary Transformations......Page 192.djvu 9.2. Continuous Spacetime Transformations in Quantum Mechanics......Page 195.djvu 9.3. Epilogue......Page 200.djvu Appendixes......Page 205.djvu A. Scalars, Vectors, Tensors, and Coordinate Transformations......Page 205.djvu B. Special Relativity......Page 211.djvu C. Equations of Motion in Quantum Mechanics......Page 217.djvu D. Legendre Transformations and Conjugate Variables......Page 221.djvu E. The Jacobian......Page 225.djvu Bibliography......Page 229.djvu Index......Page 235.djvu Caveat......Page K 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. "This book is the outgrowth of a course taught to residents in radiation oncology at Wayne State University, at the suggestion of residents who saw a need for a technically accurate text set at the correct mathematical level. It is intended to be a book to learn from, not a comprehensive compendium. It is written for members of the radiation therapy community such as radiation therapy technologists, dosimetrists, and radiation oncologists who may have taken college physics several years previously but still need to know the basic physics of radiation therapy. For graduate students in medical physics, it will serve as a review of the 'basics.' The material is written to be relevant to clinical practice, without covering specifics in treatment planning, and also with a close eye on board certification requirements."--Publisher description "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."--Publisher's description An introductory textbook to the physics and technology used in radiation therapy that is the outgrowth of a course taught to medical residents in radiation oncology and which has been classroom tested over many years. The first half of the book contains the radiation physics necessary to understand radiation therapy. The second half covers the applied physics and technology of radiation therapy. 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 title presents an introduction to the theorem's genesis, applications, and consequences that unpacks its universal importance and unsurpassed elegance