Contents

Acknowledgments

1 Lagrangian Mechanics

1.1 Configuration Spaces

1.2 Generalized Coordinates

1.3 The Principle of Stationary Action

1.4 Computing Actions

1.5 The Euler–Lagrange Equations

1.5.1 Derivation of the Lagrange Equations

1.5.2 Computing Lagrange’s Equations

1.6 How to Find Lagrangians

1.6.1 Coordinate Transformations

1.6.2 Systems with Rigid Constraints

1.6.3 Constraints as Coordinate Transformations

1.6.4 The Lagrangian Is Not Unique

1.7 Evolution of Dynamical State

1.8 Conserved Quantities

1.8.1 Conserved Momenta

1.8.2 Energy Conservation

1.8.3 Central Forces in Three Dimensions

1.8.4 The Restricted Three-Body Problem

1.8.5 Noether’s Theorem

1.9 Abstraction of Path Functions

1.10 Constrained Motion

1.10.1 Coordinate Constraints

1.10.2 Derivative Constraints

1.10.3 Nonholonomic Systems

2 Rigid Bodies

2.1 Rotational Kinetic Energy

2.2 Kinematics of Rotation

2.3 Moments of Inertia

2.4 Inertia Tensor

2.5 Principal Moments of Inertia

2.6 Vector Angular Momentum

2.7 Euler Angles

2.8 Motion of a Free Rigid Body

2.8.1 Computing the Motion of Free Rigid Bodies

2.8.2 Qualitative Features

2.9 Euler’s Equations

2.10 Axisymmetric Tops

2.11 Spin-Orbit Coupling

2.11.1 Development of the Potential Energy

2.11.2 Rotation of the Moon and Hyperion

2.11.3 Spin-Orbit Resonances

2.12 Nonsingular Coordinates and Quaternions

2.12.1 Motion in Terms of Quaternions

3 Hamiltonian Mechanics

3.1 Hamilton’s Equations

3.1.1 The Legendre Transformation

3.1.2 Hamilton’s Equations from the Action Principle

3.1.3 A Wiring Diagram

3.2 Poisson Brackets

3.3 One Degree of Freedom

3.4 Phase Space Reduction

3.4.1 Lagrangian Reduction

3.5 Phase Space Evolution

3.5.1 Phase-Space Description Is Not Unique

3.6 Surfaces of Section

3.6.1 Periodically Driven Systems

3.6.2 Computing Stroboscopic Surfaces of Section

3.6.3 Autonomous Systems

3.6.4 Computing Hénon–Heiles Surfaces of Section

3.6.5 Non-Axisymmetric Top

3.7 Exponential Divergence

3.8 Liouville’s Theorem

3.9 Standard Map

4 Phase Space Structure

4.1 Emergence of the Divided Phase Space

4.2 Linear Stability

4.2.1 Equilibria of Differential Equations

4.2.2 Fixed Points of Maps

4.2.3 Relations Among Exponents

4.3 Homoclinic Tangle

4.3.1 Computation of Stable and Unstable Manifolds

4.4 Integrable Systems

4.5 Poincaré–Birkhoff Theorem

4.5.1 Computing the Poincaré–Birkhoff Construction

4.6 Invariant Curves

4.6.1 Finding Invariant Curves

4.6.2 Dissolution of Invariant Curves

5 Canonical Transformations

5.1 Point Transformations

5.2 General Canonical Transformations

5.2.1 Time-Dependent Transformations

5.2.2 Abstracting the Canonical Condition

5.3 Invariants of Canonical Transformations

5.4 Generating Functions

5.4.1 F₁ Generates Canonical Transformations

5.4.2 Generating Functions and Integral Invariants

5.4.3 Types of Generating Functions

5.4.4 Point Transformations

5.4.5 Total Time Derivatives

5.5 Extended Phase Space

5.5.1 Poincaré–Cartan Integral Invariant

5.6 Reduced Phase Space

6 Canonical Evolution

6.1 Hamilton–Jacobi Equation

6.1.1 Harmonic Oscillator

6.1.2 Hamilton–Jacobi Solution of the Kepler Problem

6.1.3 F₂ and the Lagrangian

6.1.4 The Action Generates Time Evolution

6.2 Time Evolution is Canonical

6.2.1 Another View of Time Evolution

6.2.2 Yet Another View of Time Evolution

6.3 Lie Transforms

6.5 Exponential Identities

7 Canonical Perturbation Theory

7.1 Perturbation Theory with Lie Series

7.2 Pendulum as a Perturbed Rotor

7.2.1 Higher Order

7.2.2 Eliminating Secular Terms

7.3 Many Degrees of Freedom

7.3.1 Driven Pendulum as a Perturbed Rotor

7.4 Nonlinear Resonance

7.4.1 Pendulum Approximation

7.4.2 Reading the Hamiltonian

7.4.3 Resonance-Overlap Criterion

7.4.4 Higher-Order Perturbation Theory

7.4.5 Stability of the Inverted Vertical Equilibrium

8 Appendix: Scheme

9 Appendix: Our Notation

List of Exercises