Contents › Part 3

Part 3 — Classical Mechanics

1. Kinematics
   • Foundations
   • Galilean relativity
   • Circular motion
2. Newtonian Dynamics
   • Inertia and mass
   • Newton's laws and the Newtonian formulation: translation
   • Newton's laws and the Newtonian formulation: rotation
   • Energy
   • Systems of particles
3. Experiment: Free Fall and Projectile Motion experiment
   • Historical context: the Aristotelian expectation and its collapse
   • The observed phenomenon and its derivation
   • Galileo's Inclined Planes (1604–1638)
   • Absolute Ballistic Gravimetry (1995–present)
   • Deviations and the domain of validity
   • From Galileo's ratios to the equivalence principle
4. Lagrangian Mechanics
   • Generalized coordinates
   • Virtual work
   • d'Alembert's principle
   • The Euler–Lagrange equations
   • Hamilton's principle
   • Nonholonomic constraints
   • General coordinate transformations
   • The universality of general covariance
5. Hamiltonian Mechanics
   • From the Lagrangian to the Hamiltonian
   • Canonical transformations
   • Invariant integrals
   • Motion in phase space
   • Symplectic formulation
   • Beyond the canonical formalism
6. Hamilton–Jacobi Theory and the Optical–Mechanical Analogy
   • The Hamilton–Jacobi equation
   • Complete integrals and separation of variables
   • Action–angle variables and integrable systems
   • The optical–mechanical analogy
   • From Hamilton–Jacobi to the Schrödinger equation
7. Symplectic Geometry of Phase Space
   • Symplectic vector spaces
   • Symplectic manifolds
   • Hamiltonian vector fields and the flow
   • Invariants of the flow
   • Poisson manifolds
   • Symmetry, momentum maps, and reduction
   • Geometric quantization
   • What the symplectic view buys
8. The Poisson Algebra and the Canonical Bridge to Quantum Mechanics
   • The Poisson algebra of observables
   • Generators, symmetries, and constants of motion
   • The first-order formalism
   • Canonical quantization: the correspondence rule
   • The obstruction: what the correspondence cannot be
   • What carries over to field theory
9. Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism
   • Singular Lagrangians and primary constraints
   • The Dirac–Bergmann consistency algorithm
   • First- and second-class constraints
   • The Dirac bracket
   • Gauge fixing and the reduced phase space
   • Worked systems
   • Quantizing a constrained system
   • Constraints in field theory
10. Central Forces and Statics
    • Statics
    • Central forces
    • The equivalent one-dimensional problem
    • The two-body problem and Kepler orbits
    • The virial theorem
    • Where the claims of this chapter are tested
11. Oscillations and Mechanical Waves
    • The harmonic oscillator
    • Damped and driven oscillations
    • Coupled oscillators and normal modes
    • The vibrating string and the wave equation
    • Sound
    • Dispersion
    • Anharmonicity: a first look
12. Rigid Bodies and Rotating Frames
    • Rigid-body kinematics
    • The inertia tensor
    • Euler's equations and free rotation
    • The heavy top
    • Motion in rotating frames
13. Continuum Mechanics and Elasticity
    • The continuum hypothesis
    • Deformation: the strain tensor
    • Stress: the Cauchy tensor
    • Linear elasticity: Hooke's law and the moduli
    • Elastodynamics and elastic waves
    • Rods, beams and plates
    • Static problems and contact
    • The limits of elasticity
14. Fluid Dynamics
    • Kinematics of flow
    • Hydrostatics
    • Ideal flow: the Euler equations
    • Viscous flow: the Navier–Stokes equations
    • Dynamical similarity
    • Vorticity
    • Lift and the boundary layer
    • Instability and transition
    • Turbulence
    • Compressible flow
    • Surface waves and solitary waves
15. Nonlinear Dynamics and Chaos
    • Flows in phase space
    • Limit cycles
    • Poincaré sections and discrete dynamics
    • Hamiltonian chaos
    • Strange attractors
    • Universal routes to chaos
    • Experimental chaos
16. Experiment: The Cavendish Torsion Balance experiment
    • Historical context and the prediction under test
    • Torsion pendulum theory
    • Apparatus
    • Procedure
    • Observations and data
    • Interpretation
    • Modern determinations and their discrepancies
    • Primary references
17. Experiment: The Pendulum experiment
    • Galileo and Huygens: the isochronous pendulum (1602–1673)
    • Coulomb: the torsion pendulum (1784)
    • Kater: the reversible pendulum (1818)
    • The seconds pendulum and the metre
    • Foucault: the pendulum at the Panthéon (1851)
    • The pendulum as the first precision instrument
18. Experiment: Waves and Acoustics experiment
    • Mersenne: the laws of the vibrating string (1636)
    • Melde: standing waves on a driven string (1860)
    • Chladni: the nodal figures of a vibrating plate (1787)
    • Newton, Laplace and the speed of sound (1687–1822)
    • Kundt's tube: sound speeds from dust figures (1866)
    • Buys Ballot: the Doppler effect on a moving train (1845)
    • Summary of the evidence
19. Experiment: Fluid Flow and Turbulence experiment
    • Efflux from a vessel: Torricelli's law (1738)
    • Poiseuille: flow in fine capillaries (1840)
    • Reynolds: the dye filament and the critical number (1883)
    • Bénard: cellular convection (1900)
    • Prandtl: the boundary layer made visible (1904)
    • What the record establishes, and what it does not