Emmy Noether's wonderful theorem

Preface

Acknowledgments

Glossary of Symbols

Primary and Auxiliary Questions

1. Prologue

1.1. Symmetry, Invariance, and Conservation Laws

1.2. Emmy Noether Biographical Notes

Part I. When Functionals Are Extremal

2. Functionals

2.1. Single-Integral Functionals

2.2. Formal Definition of a Functional

3. Extremals

3.1. The Euler-Lagrange Equation

3.2. Corollaries to the Euler-Lagrange Equation

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

3.4. Where Did Hamilton's Principle Come From?

3.5. Why Kinetic Minus Potential Energy?

3.6. Extremals with External Constraints

Part II. When Functionals Are Invariant

4. Invariance

4.1. Formal Definition of Invariance

4.2. Condition for Invariance: The Rund-Trautman Identity

4.3. A More Liberal Definition of Invariance

5. Emmy Noether's Elegant Theorem

5.1. Extremal + Invariance = Noether's Theorem

5.2. The Inverse Problem: Finding Invariances

5.3. Adiabatic Invariance and Noether's Theorem

Part III. The Invariance of Fields

6. Fields and Noether's Theorem

6.1. Multiple-Integral Functionals

6.2. Euler-Lagrange Equations for Fields

6.3. Canonical Momenta and the Hamiltonian for Fields

6.4. Equations of Continuity

6.5. The Rund-Trautman Identity for Fields

6.6. Noether's Theorem for Fields

6.7. Complex Fields

6.8. Global Gauge Transformations

7. Gauge Invariance as a Dynamical Principle

7.1. Local Gauge Invariance and the Covariant Derivative

7.2. Electrodynamics as a Gauge Theory I: Field Tensors

7.3. Pure Electrodynamics, Spacetime Invariances, and Conservation Laws

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

7.5. Internal Degrees of Freedom

7.6. Non-Abelian Gauge Transformations

Part IV. Post-Noether Invariance

8. Invariance in Phase Space

8.1. Phase Space

8.2. Hamilton's Principle in Phase Space

8.3. Noether's Theorem through Hamilton's Equations

8.4. Hamilton-Jacobi Theory

9. The Action as a Generator

9.1. Conservation of Probability and Unitary Transformations

9.2. Continuous Spacetime Transformations in Quantum Mechanics

9.3. Epilogue

Appendixes

A. Scalars, Vectors, Tensors, and Coordinate Transformations

B. Special Relativity

C. Equations of Motion in Quantum Mechanics

D. Legendre Transformations and Conjugate Variables

E. The Jacobian

Bibliography

Index