Emmy Noether's Wonderful Theorem

Preface
Acknowledgments
Questions
I. WHEN FUNCTIONALS ARE EXTREMAL
1. Symmetry
1.1. Symmetry, Invariances, and Conservation Laws
1.2. Meet Emmy Noether
2. Functionals
2.1. Single-Integral Functionals
2.2. Formal Definition of a Functional
3. Extremals
3.1. The Euler-Lagrange Equation
3.2. Conservation Laws as Corollariesto the Euler-Lagrange Equation
3.3. On the Equivalence of Hamilton's Principleand Newton's Second Law
3.4. Where Do Functional Extremal PrinciplesCome From?
3.5. Why Kinetic Minus Potential Energy?
3.6. Extremals with External Constraints
II. WHEN FUNCTIONALS ARE INVARIANT
4. Invariance
4.1. Formal Definition of Invariance
4.2. The Invariance Identity
4.3. A More Liberal Definition of Invariance
5. Emmy Noether's Elegant (First) Theorem
5.1. Invariance + Extremal = Noether's Theorem
5.2. Executive Summary of Noether's Theorem
5.3. ""Extremal"" or ""Stationary""?
5.4. An Inverse Problem
5.5. Adiabatic Invariance in Noether's Theorem
III. THE INVARIANCE OF FIELDS
6. Noether's Theorem and Fields
6.1. Multiple-Integral Functionals
6.2. Euler-Lagrange Equations for Fields
6.3. Canonical Momentum and the HamiltonianTensor for Fields
6.4. Equations of Continuity
6.5. The Invariance Identity for Fields
6.6. Noether's Theorem for Fields
6.7. Complex Fields
6.8. Global Gauge Transformations
7. Local Gauge Transformations of Fields
7.1. Local Gauge Invariance and Minimal Coupling
7.2. Electrodynamics as a Gauge Theory,Part 1
7.3. Pure Electrodynamics, Spacetime Invariances,and Conservation Laws
7.4. Electrodynamics as a Gauge Theory,Part 2
7.5. Local Gauge Invariance and Noether Currents
7.6. Internal Degrees of Freedom
7.7. Noether's Theorem and GaugedInternal Symmetries
8. Emmy Noether's Elegant (Second) Theorem
8.1. Two Noether Theorems
8.2. Noether's Second Theorem
8.3. Parametric Invariance
8.4. Free Fall in a Gravitational Field
8.5. The Gravitational Field Equations
8.6. The Functionals of General Relativity
8.7. Gauge Transformations on Spacetime
8.8. Noether's Resolution of an Enigma inGeneral Relativity
IV. TRANS-NOETHER INVARIANCE
9. Invariance in Phase Space
9.1. Phase Space
9.2. Hamilton's Principle in Phase Space
9.3. Noether's Theorem and Hamilton's Equations
9.4. Hamilton-Jacobi Theory
10. The Action as a Generator
10.1. Conservation of Probabilityand Continuous Transformations
10.2. The Poetry of Nature
Appendixes
A. Scalars, Vectors, and Tensors
B. Special Relativity
C. Equations of Motion in Quantum Mechanics
D. Conjugate Variables and Legendre Transformations
E. The Jacobian
F. The Covariant Derivative
Bibliography
Index