| 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
|
