- Title Pages
- Dedication
- Preface to Second Edition
- Preface to First Edition
- Acknowledgments
- 1 Basic Dynamics of Point Particles and Collections
- 2 Introduction to Lagrangian Mechanics
- 3 Lagrangian Theory of Constraints
- 4 Introduction to Hamiltonian Mechanics
- 5 The Calculus of Variations
- 6 Hamilton's Principle
- 7 Linear Operators and Dyadics
- 8 Kinematics of Rotation
- 9 Rotational Dynamics
- 10 Small Vibrations about Equilibrium
- 11 Central Force Motion
- 12 Scattering
- 13 Lagrangian Mechanics with Time as a Coordinate
- 14 Hamiltonian Mechanics with Time as a Coordinate
- 15 Hamilton'S Principle and Noether's Theorem
- 16 Relativity and Spacetime
- 17 Fourvectors and Operators
- 18 Relativistic Mechanics
- 19 Canonical Transformations
- 20 Generating Functions
- 21 Hamilton-Jacobi Therory
- 22 Angle‐Action Variables
- Appendix A Vector Fundamentals
- Appendix B Matrices and Determinants
- Appendix C Eigenvalue Problem with General Metric
- Appendix D The Calculus of Many Variables
- Appendix E Geometry of Phase Space
- References
- Index

# Angle‐Action Variables

# Angle‐Action Variables

- Chapter:
- (p.509) 22 Angle‐Action Variables
- Source:
- Analytical Mechanics for Relativity and Quantum Mechanics
- Author(s):
### Oliver Davis Johns

- Publisher:
- Oxford University Press

This chapter introduces the angle-action method and provides several examples of its use, including its essential role in the ‘old’ quantum theory, a precursor to the Schroedinger wave equation that sought to introduce quantum ideas into a classical model. Angle-action methods are also used in what is called canonical perturbation theory, the study of systems that deviate only slightly from cyclic ones. That vast subject is best studied using the language of differential geometry. Study of the present chapter will give the reader a firm grasp of the angle-action foundation upon which canonical perturbation theory is constructed. Certain specialised systems of importance in astronomy and quantum theory can be treated by a canonical transformation to angle-action variables.

*Keywords:*
angle-action method, old quantum theory, Schroedinger wave equation, canonical perturbation theory, differential geometry

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- Title Pages
- Dedication
- Preface to Second Edition
- Preface to First Edition
- Acknowledgments
- 1 Basic Dynamics of Point Particles and Collections
- 2 Introduction to Lagrangian Mechanics
- 3 Lagrangian Theory of Constraints
- 4 Introduction to Hamiltonian Mechanics
- 5 The Calculus of Variations
- 6 Hamilton's Principle
- 7 Linear Operators and Dyadics
- 8 Kinematics of Rotation
- 9 Rotational Dynamics
- 10 Small Vibrations about Equilibrium
- 11 Central Force Motion
- 12 Scattering
- 13 Lagrangian Mechanics with Time as a Coordinate
- 14 Hamiltonian Mechanics with Time as a Coordinate
- 15 Hamilton'S Principle and Noether's Theorem
- 16 Relativity and Spacetime
- 17 Fourvectors and Operators
- 18 Relativistic Mechanics
- 19 Canonical Transformations
- 20 Generating Functions
- 21 Hamilton-Jacobi Therory
- 22 Angle‐Action Variables
- Appendix A Vector Fundamentals
- Appendix B Matrices and Determinants
- Appendix C Eigenvalue Problem with General Metric
- Appendix D The Calculus of Many Variables
- Appendix E Geometry of Phase Space
- References
- Index