The Architecture of Motion

Classical Mechanics in 22 chapters — from “Configuration Space and the Geometry of Motion” to “Stability Theory: Linearization, Floquet Theory, and Arnold Diffusion”.

Chapters

  1. Configuration Space and the Geometry of Motion
    • Degrees of Freedom and Configuration Manifolds
    • Generalised Coordinates and Constraint Surfaces
    • Velocities as Tangent Vectors
    • Time Derivatives in Generalised Coordinates
    • Examples: Pendulum, Double Pendulum, Rigid Body
    • Summary and forward view
    • Problems
    • Selected solutions
  2. Newton’s Laws and the Newtonian Program
    • The Three Laws: Axioms and Their Domain
    • Single-Particle Dynamics in Various Force Fields
    • Systems of Particles: Internal and External Forces
    • Constraints and Constraint Forces
    • The Newtonian Program’s Limitations
    • Summary and forward view
    • Problems
    • Selected solutions
  3. The Principle of Stationary Action
    • The Action Functional and Hamilton’s Principle
    • The Calculus of Variations: Deriving Euler-Lagrange
    • From Lagrangian to Newton: The Special Case L=T−V
    • Why Stationary, Not Minimum? The Role of Conjugate Points
    • Historical Interlude: Maupertuis, Euler, and Lagrange
    • Summary and forward view
    • Problems
    • Selected solutions
  4. Lagrangian Mechanics: Formalism and First Applications
    • The Lagrangian in Generalized Coordinates
    • Cyclic Coordinates and Conserved Momenta
    • The Energy Function and When It Is Conserved
    • Generalized Forces and Non-Conservative Systems
    • Worked Examples
    • Summary and forward view
    • Problems
    • Selected solutions
  5. Noether’s Theorem and the Deep Structure of Conservation
    • Continuous Transformations and Infinitesimal Generators
    • Noether’s Theorem: The Coordinate Proof
    • Noether’s Theorem: The Geometric Interpretation
    • The Ten Classical Conserved Quantities
    • Historical Note: Noether’s 1918 Paper and Its Reception
    • Summary and forward view
    • Problems
    • Selected solutions
  6. Rotational Kinematics: The Geometry of SO(3)
    • Rotations as Active Transformations, and the Group SO(3)
    • Euler's Rotation Theorem and the Axis-Angle Representation
    • Infinitesimal Rotations and the Lie Algebra so(3)
    • Angular Velocity and the Transport Theorem
    • Euler Angles and the Gimbal Singularity
    • The Topology of SO(3): Real Projective Space and the Plate Trick
    • Summary and forward view
    • Problems
    • Selected solutions
  7. Rigid Body Dynamics I: Inertia Tensor and Euler’s Equations
    • Kinematics of Rigid Bodies: Configuration Space SO(3)×R3
    • The Inertia Tensor: Definition and Principal Axes
    • Angular Momentum and Kinetic Energy in Body and Space Frames
    • Euler’s Equations for Torque-Free Motion
    • The Poinsot Construction and Geometric Visualization
    • Stability of Rotation About Principal Axes
    • Summary and forward view
    • Problems
    • Selected solutions
  8. Rigid Body Dynamics II: The Heavy Symmetric Top
    • Euler Angles and the Kinetic Energy of a Symmetric Body
    • The Lagrangian, the Cyclic Coordinates and the Two Conserved Momenta
    • Reduction to One Dimension: The Effective Potential
    • The Three Motions of the Axis: Loops, Cusps and Unidirectional Precession
    • Steady Precession, Its Two Roots, and the Fast-Top Limit
    • The Sleeping Top, and Where This Treatment Breaks Down
    • Summary and forward view
    • Problems
    • Selected solutions
  9. Small Oscillations and the Eigenvalue Problem
    • Equilibrium Points and Linearization
    • The Eigenvalue Problem for Coupled Oscillators
    • Normal Modes and Normal Coordinates
    • Degeneracy, Symmetry, and Mode Selection
    • Worked Examples
    • Summary and forward view
    • Problems
    • Selected solutions
  10. Driven and Damped Oscillations: Resonance and Response
    • Damping Mechanisms and the Damped Harmonic Oscillator
    • Driven Oscillations and the Steady-State Response
    • Transient Behaviour and the Green’s Function
    • Coupled Driven Oscillators: Impedance and Mode Mixing
    • Computational Exercise: Frequency Sweep and Resonance Curves
    • Summary and forward view
    • Problems
    • Selected solutions
  11. Perturbation Theory and Secular Effects
    • Time-Independent Perturbation Theory: Shifting a Normal Mode
    • The Naive Expansion and the Birth of Secular Terms
    • The Anharmonic Oscillator and the Poincaré-Lindstedt Method
    • The Method of Multiple Scales
    • Non-Conservative Perturbations: Where Damping Comes From, and the van der Pol Limit Cycle
    • Slowly Varying Parameters and the Adiabatic Invariant
    • Summary and forward view
    • Problems
    • Selected solutions
  12. The Hamiltonian Formulation: Phase Space and Canonical Equations
    • The Legendre Transformation and Conjugate Momenta
    • Hamilton's Equations and Phase Space
    • Why Hamiltonian? Advantages Over the Lagrangian Formulation
    • Poisson Brackets and the Algebra of Observables
    • Three Worked Systems: Oscillator, Central Force, Charged Particle
    • Summary and forward view
    • Problems
    • Selected solutions
  13. Symplectic Geometry and Liouville's Theorem
    • The Form on Phase Space
    • The Matrix J and the Symplectic Group
    • The Poisson Bracket Was the Symplectic Form All Along
    • Liouville's Theorem, Proved Twice
    • The Poincaré Integral Invariants
    • What Volume Conservation Forbids, and What It Forces
    • Summary and forward view
    • Selected solutions
  14. Canonical Transformations and Generating Functions
    • The Freedom of Coordinates in Phase Space
    • What Canonical Means: Two Definitions and Their Equivalence
    • Generating Functions and the Four Types
    • Four Transformations Worked Through
    • Infinitesimal Canonical Transformations and Their Generators
    • The Canonical Group
    • Summary and forward view
    • Problems
    • Selected solutions
  15. Hamilton-Jacobi Theory: Action as a Function
    • Hamilton's Principal Function and the H-J Equation
    • The H-J Equation as a Canonical Transformation to Equilibrium
    • Separation of Variables and Ignorable Coordinates
    • Worked Example: The Kepler Problem in Spherical Coordinates
    • Hamilton's Characteristic Function and Time-Independent Systems
    • Connection to Quantum Mechanics: The Semiclassical Limit
    • Summary and forward view
    • Problems
    • Selected solutions
  16. Integrability and the Liouville-Arnold Theorem
    • Constants of Motion and Involution
    • Complete Integrability: The Liouville Condition
    • The Liouville-Arnold Theorem and Torus Foliation
    • Action-Angle Variables: Definition and Construction
    • Frequencies and the Frequency Map
    • Summary and forward view
    • Problems
    • Selected solutions
  17. Central Force Motion and Orbital Mechanics
    • Reduction to the One-Body Problem and Effective Potential
    • The Orbit Equation and Kepler's First Law
    • Kepler's Second and Third Laws from Conserved Quantities
    • The Laplace-Runge-Lenz Vector and SO(4) Symmetry
    • Action-Angle Variables for the Kepler Problem
    • Perturbations and Orbital Precession
    • Summary and forward view
    • Problems
    • Selected solutions
  18. The Three-Body Problem and the Onset of Chaos
    • The Restricted Circular Three-Body Problem
    • The Jacobi Integral and Zero-Velocity Curves
    • The Five Lagrange Points and Their Stability
    • Poincaré Sections and the Breakdown of Integrability
    • Computational Exercise: Simulating the Three-Body Problem
    • Summary and forward view
    • Problems
    • Selected solutions
  19. Perturbation Theory for Nearly Integrable Systems
    • The Nearly Integrable Setting
    • First-Order Canonical Perturbation Theory
    • Secular and Oscillatory: The Averaging Principle
    • Resonance and the Resonant Normal Form
    • The Pendulum Approximation and the Resonance Width
    • Small Divisors, Divergence, and Resonance Overlap
    • Summary and forward view
    • Problems
    • Selected solutions
  20. KAM Theory and the Persistence of Invariant Tori
    • The Question KAM Answers, and Why It Was Thought Hopeless
    • Nondegeneracy, and What “Irrational Enough” Means
    • The Theorem, and What It Does Not Say
    • The Standard Map: A Laboratory for Torus Destruction
    • The Last Torus: The Golden Ratio and Greene's Criterion
    • Confinement in Two Degrees of Freedom, and Its Failure Above Two
    • Summary and forward view
    • Problems
    • Selected solutions
  21. Chaos, Lyapunov Exponents, and Sensitive Dependence
    • Sensitive Dependence and the Definition of Chaos
    • Lyapunov Exponents: Definition and Computation
    • The Lyapunov Spectrum for Higher-Dimensional Systems
    • The Driven Pendulum and the Hénon-Heiles System
    • A Computational Recipe
    • Summary and forward view
    • Problems
    • Selected solutions
  22. Stability Theory: Linearization, Floquet Theory, and Arnold Diffusion
    • What Stability Means, and the Several Things the Word Is Used For
    • Linearisation About an Equilibrium
    • Periodic Orbits, the Monodromy Matrix, and Floquet Multipliers
    • Poincaré Sections and the Reduction to a Map
    • Arnold Diffusion and the Nekhoroshev Estimate
    • What Classical Mechanics Settled, and What It Left Open
    • Summary and forward view
    • Problems
    • Selected solutions