Physics for Mathematicians, Mechanics I. Author: Michael Spivak. Publisher: Publish or Perish, Inc., Houston, TX, USA. http://www.mathpop.com/index.htm. ISBN-13: 978-0914098324. xvi + 733 pages. 9.3 x 6.6 x 1.8 inches. Hardcover, clothbound. First edition, 2010. Rating: **** (good)

• Preface
• 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
    • 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
    • Addendum 3B. Follow the Bouncing Ball
    • 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. A 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 see pages 246, 247, 248, 249
    • Addendum 6A. The Bouncing SuperBall see pages 250, 251
    • 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
    • Couple oscillators
    • The double pendulum
    • The vibrating string see pages 310, 311, 312, 313
    • 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 Coriolise 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 Jellet 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 see pages 444, 445
    • Constraint problems see pages 446, 447
    • Conservation of energy; action see pages 448, 449
    • 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 Volum 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 SCHRODINGER 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
    • Condition 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. A PDE Primer
• Bibliography
  • Unabbreviated Journal Titles
• Index

    Physics is revisted by a very good professor of mathematics.

    In 2004, he wrote: "the problem is that I could easily understand symplectic structures, it's elementary mechanics that I don't understand". Then he worked hard, and produced this wonderful summa.

    If learning mechanics is also your challenge, then you will profit from the valuable knowledge gathered and arranged by Michael Spivak.