8.09 | Fall 2014 | Undergraduate

Classical Mechanics III

Lecture Notes

The full course notes (PDF - 6.1MB) are also available for download.

1 A Review of Analytical Mechanics (PDF)

  • Lagrangian & Hamiltonian Mechanics
  • Symmetry and Conservation Laws
  • Constraints and Friction Forces
  • Calculus of Variations & Lagrange Multipliers
2 Rigid Body Dynamics (PDF)

  • Coordinates of a Rigid Body
  • Time Evolution with Rotating Coordinates
  • Kinetic Energy, Angular Momentum, and the Moment of Inertia Tensor for Rigid Bodies
  • Euler Equations
  • Symmetric Top with One Point Fixed
3 Vibrations & Oscillations (PDF)

  • Simultaneous Diagonalization of T and V
  • Vibrations and Oscillations with Normal Coordinates
4 Canonical Transformations, Hamilton-Jacobi Equations, and Action-Angle Variables (PDF)

  • Generating Functions for Canonical Transformations
  • Poisson Brackets and the Symplectic Condition
  • Equations of Motion & Conservation Theorems
  • Hamilton-Jacobi Equation
  • Kepler Problem
  • Action-Angle Variables
5 Perturbation Theory (PDF)

  • Time Dependent Perturbation Theory for the Hamilton-Jacobi Equations
  • Periodic and Secular Perturbations to Finite Angle Pendulum
  • Perihelion Precession from Perturbing a Kepler Orbit
6 Fluid Mechanics (PDF)

  • Transitioning from Discrete Particles to the Continuum
  • Fluid Equations of Motion: Continuity Equations, Ideal Fluid: Euler’s Equation and Entropy Conservation, Conservation of Momentum and Energy
  • Static Fluids & Steady Flows
  • Potential Flow
  • Sound Waves
  • Viscous Fluid Equations
  • Viscous Flows in Pipes and Reynolds Number
  • Viscous Flow Past a Sphere (Stokes Flow)
7 Chaos and Non-Linear Dynamics (PDF - 4.8MB)

  • Introduction to Chaos: Evolution of the System by First Order Differential Equations, Evolution of Phase Space, Fixed Points, Picturing Trajectories in Phase Space
  • Bifurcations: Saddle-Node Bifurcation, Transcritical Bifurcation, Supercritical Pitchfork Bifurcation, Subcritical pitchfork bifurcation
  • Fixed Points in Two-Dimensional Systems: Motion Near a Fixed Point, Systems with a conserved E(x)
  • Limit Cycles and Bifurcations: Poincare-Bendixson Theorem, Fixed Point Bifurcations Revisited and Hopf Bifurcations
  • Chaos in Maps
  • Chaos in Differential Equations, Strange Attractors, and Fractals: The Lorenz Equations, Fractals and the Connection to Lyapunov Exponents, Chaos in Fluids

Course Info

As Taught In
Fall 2014
Learning Resource Types
Problem Sets
Lecture Notes