Lecture Notes:
The lecture notes are based on several sources
- Our text book -- Landau.
- The standard text -- Goldstein, Poole, Safko
- Professor David Tong's lecture notes are on point.
- Professor Likharev gives a superb discussion of oscillations.
- Bender and Orzag, Advanced Mathematical Methods for Scientists and Engineers , is a classic reference on secular perturbation theory.
- Professor Fowler's notes gives helpful guidance to Landau.
Video Recordings of Lectures:
Video recordings of the lectures are available.
Outline:
- Basic mechanics
- Constraints and symmetries in mechanics
- Rigid Body Motion and Accelerating Frames
- Oscillations
- More about Hamiltonian mechanics
- The Hamilton Jacobi Formulation
- Waves
- Newton laws:
- The action principle and lagrange equations of motion: ,
- Hamilton's equation of motion and the variational principle:
- Keplerian Orbits:
- Scattering:
- References:
- Motion in one dimension is treated by Landau: 11, Tong 2.6
- Newtonian and Langrangian mechanics: Landau 1, 2, 40; Tong: 1.2, 1.3, 2.1, 2.2; Goldstein: 1.1 - 1.2, Goldstein: 2.1 - 2.3
- Hamiltonian Mechanics: Landau 40; Tong: 4.1 - 4.1.3; Goldstein 8.1.
- A good introduction the Legendre transform is here
- The Kepler system is discussed in Goldstein: 3.7, 3.8, 3.9. And our notation follows his.
- Scattering follows Goldstein: 3.10
- Mechanics with constraints:
- Noether theorem:
- References:
- Constraints are nicely discussed in Goldstein 2.4.
- Noether theorem Tong 2.4.1; A good introduction to Noether is given here
- Kinematics of rotating frames
- Dynamics of rigid bodies
- Moment of Inertia, Energy, Angular Momentum
- The Euler equations and the wobble of the plate
- Euler equations for free motion:
- Watch the experiment
- Watch the simulation movies
- The symmetric top and forced motion
- References
- Our notation largely follows, Tong, Sec 3.1, (but not 3.1.2), 3.2, 3.3, 3.5, 3.6
- Likharev, 6.6, gives a good discussion of accelerating frames.
- Oscillations and Normal Modes:
- Forced Oscillations:
- The retarded Green function for the damped oscillator
- Non linear oscillations
- The pondermotive force and dynamics in rapdily oscillating force field
- Slides for this section all slides
- References:
- Vibrations of molecules is from Landau: 24
- The normal modes example is from Tong: Section 2.6.2
- The discussion of forced oscillations: Likharev 4.1, Landau 22
- Green functions are described in Likharev: 4.1
- Nonlinear oscillations: Likharev 4.2, Landau 28.
- Nonlinear resonant oscillations: Likharev 4.2, Landau 29.
- Secular perturbation theory: Bender Orzag, Chapter 11.1, 11.2
- Phase space and the Liouville theorem
- Phase space density.
- Poisson brackets
- Infinitessimal canonical transformations, and Noether's theorem revisited
- A short summary
- Adiabatic invariance and assoicated problems
- General canonical transformations
- Symplectic Integrators
- References:
- Starts by following Tong.
- Then switches to Goldstein for the most part
- Adiabatic invariance follows, Likharev and Landau.
- Symplectic integrators follow Ruth.
- The onshell action
- The Hamilton-Jacobi equation
- The Kepler Problem with the Hamilton-Jacobi formulation
Basic Mechanics:
Constraints and symmetries in Lagrangian mechanics:
Rigid body motion and accelerating frames:
Oscillations
Hamiltonian Mechanics:
The Hamilton-Jacobi Formulation:
Waves: