Action and Symmetry

Classical Dynamics after Newton in 17 chapters — from “The action, the first variation and the boundary term” to “Adiabatic invariants, canonical perturbation theory and integrable systems”.

Chapters

  1. The action, the first variation and the boundary term
    • Paths, functionals and the action
    • The first variation for n variables
    • The boundary term at a free end
    • Gauge freedom of the Lagrangian
    • The Beltrami first integral
    • Functionals outside mechanics
    • Conjugate points and the soap film
    • Problems
  2. Constraints, generalised coordinates and Lagrange's equations twice over
    • Configuration space and degrees of freedom
    • Virtual displacements and ideal constraints
    • d'Alembert's principle and the generalised force
    • Equivalence of the variational and virtual-work routes
    • Form invariance under coordinate changes
    • d'Alembert's principle on a moving wire with drag
    • Problems
  3. Multipliers, non-holonomic constraints and forces outside the potential
    • Lagrange multipliers for constrained stationarity
    • The multiplier as a constraint force
    • Constraint release from a sphere
    • Non-holonomic constraints and the integrability test
    • Multipliers for a non-holonomic constraint
    • The Rayleigh dissipation function
    • The vector potential and a generalised potential
    • Problems
  4. Symmetry, Noether's theorem and what is conserved
    • Noether's theorem up to a total derivative
    • Boost symmetry and the centre of mass
    • Jacobi integral versus the energy
    • Conservation under dissipation
    • Mechanical similarity and the virial relation
    • The Runge-Lenz vector
    • Problems
  5. Non-inertial frames from one Lagrangian
    • The Lagrangian in a moving frame
    • The Euler force
    • Jacobi integral of a rotating frame
    • Index notation and Cartesian tensors
    • The tidal tensor
    • Local equivalence of acceleration and gravity
    • Problems
  6. The inertia tensor, principal axes and Euler angles
    • The inertia tensor
    • Product of inertia and dynamic balance
    • Tensor parallel-axis theorem
    • Principal axes and the inertia ellipsoid
    • Euler angles and the angular velocity
    • Kinetic energy in Euler angles
    • Problems
  7. Euler's equations, free precession and the heavy symmetric top
    • Euler's equations
    • Free precession of a symmetric body
    • Body and space cones
    • The intermediate-axis instability
    • Effective potential of the heavy top
    • Steady precession and nutation
    • The sleeping top
    • Problems
  8. The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector
    • Reduction by a cyclic coordinate
    • The orbit by quadrature
    • The apsidal angle
    • Bertrand's theorem
    • The orbit from the Runge-Lenz vector
    • The virial theorem
    • Problems
  9. Scattering in a central field
    • Impact parameter
    • The deflection function
    • The differential cross-section
    • The Rutherford cross-section
    • The total cross-section
    • Rainbow and glory scattering
    • The laboratory-frame cross-section
    • Problems
  10. Perturbed orbits, precession and the three-body problem
    • Perturbation series to first order
    • Apsidal precession from a small potential
    • Osculating elements and slow drift
    • Quadrupole precession of satellite orbits
    • Mercury and its apsidal precession
    • Restricted three-body problem and zero-velocity curves
    • Lagrange points and their stability
    • Mean-motion resonance and small denominators
    • Problems
  11. Normal modes as a generalised eigenvalue problem
    • Kinetic and potential matrices
    • The generalised eigenvalue problem
    • Reality of eigenvalues and kinetic-matrix orthogonality
    • Normal coordinates for n degrees of freedom
    • Mode-by-mode forced response
    • Problems
  12. Small oscillations: stability, degeneracy, symmetry and spectra
    • Stability from the Hessian
    • Zero modes and continuous symmetry
    • Symmetry-forced degeneracy
    • Symmetry-adapted coordinates
    • The vibrational spectrum of carbon dioxide
    • The monatomic chain
    • Breakdown of the quadratic approximation
    • Problems
  13. The Hamiltonian, phase space and the variational principle in phase space
    • Legendre transform in n variables
    • Canonical momentum in a magnetic field
    • The Routhian
    • The phase-space variational principle
    • Phase flow and phase portraits
    • Jacobi integral versus the energy, in Hamiltonian form
    • Problems
  14. Poisson brackets, phase flow and Liouville's theorem
    • The Poisson bracket
    • Constants of the motion and Poisson's theorem
    • Angular momentum algebra
    • Liouville's theorem
    • Emittance
    • The Liouville equation
    • Problems
  15. Canonical transformations, generating functions and symplectic structure
    • Canonical transformations
    • The symplectic condition
    • Generating functions
    • Infinitesimal canonical transformations
    • Canonical invariants
    • Oscillator by generating function
    • Problems
  16. Hamilton-Jacobi theory and action-angle variables
    • The Hamilton-Jacobi equation
    • Hamilton's principal function
    • Separation of the Hamilton-Jacobi equation
    • The optical-mechanical analogy
    • Action variables and angle variables
    • Frequency degeneracy and the Runge-Lenz vector
    • The Bohr-Sommerfeld condition
    • Problems
  17. Adiabatic invariants, canonical perturbation theory and integrable systems
    • The adiabatic invariant
    • Adiabatic invariants in walls, mirrors and quantum rules
    • Canonical perturbation theory
    • Apsidal precession by averaging
    • Failure of the perturbation series
    • Integrable systems
    • Problems