Physics for Mathematicians, Mechanics I — Table of Contents

The complete table of contents of Physics for Mathematicians, Mechanics I by Michael Spivak (Publish or Perish, Inc.; ISBN 978-0-914098-32-4), a rigorous introduction to classical mechanics written for readers trained in modern mathematics, from Newton's laws through Lagrangian and Hamiltonian mechanics to symplectic geometry. Chapter and section headings are listed as they appear in the book.

PART I · THE FOUNDATIONS OF MECHANICS

Prologue

Chapter 1. Newtonian Mechanics

• Mass and force
• The first law
• The second law
• Mass and weight are different…
• …yet not so different
• The third law
• The lures of symmetry
• Composition of forces
• Addendum 1A. It Isn't Rocket Science (Why Easy Physics is So Hard: I)
• Addendum 1B. Weight Versus Mass
• Problems

Chapter 2. Newton's Analysis of Central Forces

• Problems

Chapter 3. Conservation Laws

• Conservation of momentum
• Conservation of angular momentum
• Conservation of energy: kinetic and potential energy
• Conservation of energy in collisions
• Conservation of energy in general
• Addendum 3A. Whips and Chains (Why Easy Physics is So Hard: II)
• Addendum 3B. Follow the Bouncing Ball (Why Easy Physics is So Hard: III)
• Problems

Chapter 4. The One-Body and Two-Body Problems

• The one-body problem
• “The motion of bodies in mobile orbits, and the motion of the absides”
• The two-body problem
• “The attractive forces of spherical bodies”
• Addendum 4A. À la Principia
• Addendum 4B. Reduction to a One-Dimensional Problem
• Addendum 4C. Rutherford Scattering
• Addendum 4D. Bertrand's Theorem
• Addendum 4E. Power Force Laws and Duality
• Problems

Chapter 5. Rigid Bodies

• Equilibrium
• Virtual infinitesimal displacements
• Configuration space
• The principle of virtual work
• d'Alembert's principle
• The inertia tensor
• Calculating the inertia tensor
• Rotation about an axis
• Kinetic energy
• Continuous bodies
• Elementary examples
• Addendum 5A. The Strong Form of the Third Law
• Problems

Chapter 6. Constraints

• Rigid bodies in contact
• The pendulum
• The compound (physical) pendulum
• Equilibrium and Stability
• Sliding
• Rolling
• Some subsidiary topics (time-dependent constraints and hinges)
• Holonomic and differential constraints
• Finding the constraint forces
• The rolling sphere
• Give a physics student enough rope problems…
• Addendum 6A. The Bouncing SuperBall
• Addendum 6B. Statically Indeterminate Problems
• Problems

Chapter 7. Philosophical and Historical Questions

• Early notions of conservation of momentum
• Huygens and Galilean Invariance
• Newton's proof of the third law
• The parallelogram law
• Newton at the hands of the scholars

PART II · BUILDING ON THE FOUNDATIONS

Chapter 8. Oscillations

• Huygens' cycloidal pendulum
• The spherical pendulum
• Springs
• Harmonic oscillations
• Damped oscillations
• Forced oscillations
• Damped forced oscillations
• Coupled oscillators
• The double pendulum
• The vibrating string
• Addendum 8A. Abel's Integral Equation
• Addendum 8B. Envelopes
• Addendum 8C. Stability of Solutions of Differential Equations
• Problems

Chapter 9. Rigid Body Motion

• Rotating coordinate systems
• The Euler equations
• Poinsot's geometric description
• The free symmetric top, in body coordinates
• The free symmetric top, in inertial coordinates
• Euler angles
• The heavy symmetrical top
• The cuspidal case; fast tops
• Precessing tops
• Sleeping tops
• The rising top
• The polar cuspidal top
• Gyroscopes
• The gyrocompass
• Precession of the equinoxes
• Addendum 9A. The Euler Equations for Rotating Principal Vectors — The Rolling Disc
• Addendum 9B. Secrets of the Herpolhode
• Problems

Chapter 10. Non-Inertial Systems and Fictitious Forces

• The basic equations
• The translational or acceleration force
• The centrifugal force
• The deflection of a hanging body
• The azimuthal or Euler force
• The Coriolis force
• The deflection of a falling body
• The southward deflection
• Stupid experimenter tricks
• Foucault's pendulum
• Hurricanes and bath-tubs
• Mach's Principle
• Addendum 10A. The Trojan Asteroids
  • The restricted three-body problem
  • Stability
  • Stability calculations
  • The collinear Lagrange points
• Addendum 10B. The Southward Deflection
• Problems

Chapter 11. Friction, Friend and Foe

• The laws of friction
• The Painlevé paradoxes
• The noble game of billiards
• The Jellett invariant
• Tippe Tops and hard boiled eggs
• Problems

PART III · LAGRANGIAN MECHANICS

Chapter 12. Analytical Mechanics

• The mathematical arena for analytical mechanics
• Specialized considerations for analytical mechanics
• Lagrange's equations
• Using Lagrange's equations
• Constraint problems
• Conservation of energy; action
• Time-dependent Lagrangians
• Lagrange multipliers
• Addendum 12A. Lagrange's Rolling Disc
• Problems

Chapter 13. Variational Principles

• The Euler equations
• Hamilton's principle
• Maupertuis and the Principle of Least Action
• Jacobi's form of the principle of least action
• Noether's Theorem
• The lures of symmetry, advanced version
• Addendum 13A. Lagrange Multipliers for Conditional Critical Points
• Problem

Chapter 14. Small Oscillations

• Problems

INTERLUDE

Chapter 15. Light

• Optics in antiquity
  • Islamic scholars
  • Kepler and Galileo
  • Descartes
  • Fermat
  • Huygens
  • Newton
  • Maupertuis
  • Malus
• Addendum 15A. Battling to a Draw
• Addendum 15B. Huygens' Principle

PART IV · HAMILTONIAN MECHANICS From Aragonite to the Schrödinger Wave Equation

Chapter 16. The Cotangent Bundle

• Special features of the cotangent bundle
• The Legendre transform
• Addendum 16A. The Clairaut Equation
• Problems

Chapter 17. The Interplay of Mechanics and Optics

• Optics emulates mechanics
  • Malus' Theorem
  • Fermat's Principle and Huygens' Construction
  • Conical Refraction in Aragonite
• Mechanics returns the compliment
  • The equations on T*M
  • The partial derivatives of S
  • A partial differential equation for S
  • Invariant definitions; the interplay of TM and T*M
  • The extended Hamilton's principle
• Addendum 17A. Liouville's Volume Theorem
• Problems

Chapter 18. Hamilton–Jacobi Theory

• The complete integral
• (Optional) Envelopes of solutions
• (Optional) Inverting the process; contact curves
• Jacobi's Theorem
• Jacobi's theorem and mechanics
• Hamilton's characteristic function
• Hamilton–Jacobi Theory and the Schrödinger Wave Equation
• Addendum 18A. Motion in the Field of Two Fixed Masses
  • Geodesics on Ellipsoids
• Addendum 18B. Huygens' Construction for Hyperbolic Equations
• Problem

Chapter 19. Canonical Transformations

• Canonical transformations
• Hamiltonian flows and integral invariants
• Hamiltonian flows and canonical transformations
• Generating functions
• Time-dependent canonical transformations
• Using generating functions to simplify Hamilton's equations
• Generating functions in the time-independent case
• Other types of generating functions
• Addendum 19A. Time-(In)Dependent Hamiltonians
• Addendum 19B. Generalized Canonical Transformations
• Problems

Chapter 20. Symplectic Manifolds

• Symplectic vector spaces
• Isotropic subspaces
• Symplectic manifolds
• Poisson brackets
• Poisson brackets bis
• Problems

Chapter 21. Liouville Integrability

• Functions in involution
• Conditional periodicity and the invariant tori
• Action-angle variables
• Action-angle variables on symplectic manifolds
• Background
• Problems

Chapter 22. Epilogue

• Adiabatic invariants
• The averaging principle
• An averaging theorem for one-dimensional systems
• Adiabatic invariance of J
• The Hannay angle
• The Hannay hoop
• Foucault's pendulum revisited
• Problems

SUPPLEMENT & BACK MATTER

Supplement. A PDE Primer

Bibliography

• Unabbreviated Journal Titles

Index

Related Reading

See also the prerequisite Comprehensive Introduction to Differential Geometry (Volumes 1 and 2 are the stated prerequisite), the complete index (A–H · I–P · Q–Z), and the biography and complete bibliography of Michael Spivak.