Analytical Mechanics

Mechanics rebuilt on a variational principle

About this course

Lagrangian and Hamiltonian mechanics from the principle of stationary action, through symmetry and conservation to phase space and chaos.

What you will be able to do

  • Derive the Euler-Lagrange equations from the principle of stationary action
  • Solve constrained mechanical problems with generalised coordinates
  • Connect symmetry to conservation via Noether's theorem
  • Move between Lagrangian and Hamiltonian descriptions and use Poisson brackets
  • Analyse small oscillations, rigid bodies and central-force orbits

What this course covers

  • Determinism versus predictability
  • Exponential error growth and the logarithmic forecast horizon
  • The largest Lyapunov exponent: definition and measurement
  • Control experiment: distinguishing physical chaos from numerical artefact
  • Integrability, action-angle variables, and the Liouville–Arnold theorem
  • KAM theorem and the transition from regular to chaotic motion
  • Scope and legacy of analytical mechanics
  • Expanding about equilibrium: two quadratic forms
  • Historical context: secular equation and Weierstrass's theorem
  • Matrix equation of motion and the generalized eigenvalue problem
  • Limits of the harmonic approximation
  • Normal modes: physical meaning and orthogonality
  • Interpreting eigenvalue signs: oscillation, zero modes, and instability
  • Worked example: collinear CO₂ vibrational modes
  • Why replace velocity with momentum
  • Phase space and its geometry
  • The Legendre transform
  • Constructing the Hamiltonian
  • Hamilton's equations of motion
  • Worked oscillator in phase space
  • Action variable and old quantum theory
  • First integrals and reduction of order
  • Cyclic coordinates and conjugate momentum conservation
  • Ignorable vs. cyclic: historical terminology
  • Kepler's second law as angular momentum conservation
  • Coordinate-choice dependence vs. physical symmetry
  • Reduction to one-dimensional effective potential
  • Conservation laws as exact physical facts
  • Energy as a first integral from time-translation invariance
  • Newtonian vs. analytical mechanics: scalars over vectors
  • Degrees of freedom: coordinates minus constraints
  • Constraint forces do no work
  • Generalized coordinates and built-in constraints
  • Configuration space and its geometry
  • Effective potential and bifurcation: bead on spinning hoop
  • Double pendulum setup: constraints as coordinate geometry
  • Continuous symmetry: precise definition
  • Noether's theorem: statement and two-line proof
  • Discrete vs. continuous symmetries
  • Extended symmetry condition: total time derivative
  • Three classical conservation laws as one theorem
  • Numerical demonstration: symmetry breaking and conservation drift
  • Noether's theorem beyond mechanics: fields, charge, and limits in GR
  • Liouville's theorem — statement and preview
  • Ensemble and phase-space density
  • Phase space as a fluid and divergence
  • Proof that Hamiltonian divergence is zero
  • Continuity equation derivation of Liouville
  • Area preserved, shape destroyed — and irreversibility
  • Liouville as the foundation of statistical mechanics
  • The inertia tensor: definition and structure
  • Products of inertia and dynamic imbalance
  • Principal axes and principal moments
  • The inertia ellipsoid and Poinsot's construction
  • Conservation laws and the intermediate-axis flip
  • Numerical rate of the intermediate-axis instability
  • Euler's equations in the body frame
  • Functionals versus ordinary functions
  • Admissible curves and stationarity
  • Derivation of the Euler-Lagrange equation
  • Extensions: variation notation, multiple functions, fundamental lemma
  • The brachistochrone: cycloid as fastest descent
  • Beltrami identity and first integrals
  • Geodesics and the soap-film catenary
  • Poisson Bracket: Definition and Fundamental Brackets
  • Historical Context: Poisson, Jacobi, and Dirac
  • Algebraic Properties of the Poisson Bracket
  • The Bracket as Coordinate-Free Structure; Lie Algebra
  • Evolution Equation and Conservation via the Bracket
  • Angular Momentum Algebra: {Lx, Ly} = Lz
  • Dirac's Quantization Rule: Scope and Failure
  • Holonomic constraints and what elimination costs
  • The Lagrange multiplier method — recipe and derivation
  • Physical meaning and units of the multiplier λ
  • Strategic use of multipliers vs. coordinate elimination
  • Bead leaving a smooth sphere — worked derivation
  • Sign of λ as a model-validity indicator
  • Non-holonomic constraints and limits of the method
  • Action as a functional over histories
  • Stationarity condition δS = 0
  • Euler-Lagrange equation of mechanics
  • Newton's second law recovered from stationary action
  • Stationary vs. least action — conjugate points
  • Reach of the action principle across all of physics
  • Action units, Planck's constant, and the quantum connection
  • Symmetries of the action and Noether's theorem
  • Earth's orbit eccentricity: circle not oval
  • Two-body to one-body reduction via reduced mass
  • Effective potential and orbit classification
  • Reduction by conservation laws: six degrees to zero
  • Orbit equation and the u = 1/r substitution
  • Orbital closure and two independent clocks
  • Bertrand's theorem: only two force laws close every bound orbit
  • Earth's orbit computed from first principles
  • Why Hamiltonian mechanics restricts coordinate changes
  • Canonical transformations: definition and criterion
  • Symplectic condition as matrix form of canonicality
  • Generating functions and the four types
  • Action-angle variables
  • Hamilton-Jacobi equation
  • Worked bracket calculations: canonical vs. non-canonical
  • Why constraint forces never appear: the Lagrangian advantage
  • The five-step Lagrangian recipe
  • Atwood machine: one coordinate for two masses
  • Pendulum on a free cart: coupled coordinates and recoil
  • Holonomic vs non-holonomic constraints
  • Cyclic coordinates and conservation laws
  • Rolling race: shape determines acceleration
  • Honest limits of the Lagrangian method

Syllabus

  1. The Variational Principle

    1. L1 · Why Newton Isn't Enough
    2. L2 · The Calculus of Variations
    3. L3 · The Principle of Stationary Action
    4. L4 · Lagrange's Equations at Work

  2. Symmetry and Conservation

    1. L5 · Cyclic Coordinates & First Integrals
    2. L6 · Noether's Theorem
    3. L7 · Constraints & Lagrange Multipliers

  3. Hamiltonian Mechanics

    1. L8 · The Legendre Transform & Hamilton's Equations
    2. L9 · Phase Space & Liouville's Theorem
    3. L10 · Poisson Brackets
    4. L11 · Canonical Transformations & Generating Functions

  4. Applications

    1. L12 · Small Oscillations & Normal Modes
    2. L13 · Rigid Body Motion
    3. L14 · Central Forces & Orbits

  5. The Limits of Prediction

    1. L15 · Chaos & the Limits of Prediction