Emmy Noether's wonderful theorem 9780801896934, 0801896932, 9780801896941, 0801896940, 1471922162, 9781930524323

Table of contents :
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