Classical Mechanics

• Syllabus
• Lectures
• Notes
• Homework
• Exams

Lecture Notes:

The lecture notes are based on several sources

1. Our text book -- Landau.
2. The standard text -- Goldstein, Poole, Safko
3. Professor David Tong's lecture notes are on point.
4. Professor Likharev gives a superb discussion of oscillations.
5. Bender and Orzag, Advanced Mathematical Methods for Scientists and Engineers , is a classic reference on secular perturbation theory.
6. Professor Fowler's notes gives helpful guidance to Landau.

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Video Recordings of Lectures:

Video recordings of the lectures are available.

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Outline:

• Basic mechanics
• Constraints and symmetries in mechanics
• Rigid Body Motion and Accelerating Frames
• Oscillations
• More about Hamiltonian mechanics
• Waves
• The Hamilton Jacobi Formulation



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Basic Mechanics:

• Newton laws:
  • Newton Laws
  • Gallilean Boosts
• The action principle and lagrange equations of motion: ,
  • The action: part 1
  • The action: part 2 -- general coordinate and an example
  • The hamiltonian function
  • Motion of 1d systems
• Hamilton's equation of motion and the variational principle:
  • Introduction to the Hamiltonian
  • Hamiltonian and the variational principle
  • The Hamiltonian and the Legendre transform
• Keplerian Orbits:
  • Kepler part 1
  • Kepler part 2
  • Movies of orbits with normal and modified gravity
• Scattering:
  • Cross section
• 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

Constraints and symmetries in Lagrangian mechanics:

• Mechanics with constraints:
  • Lagrange multipliers
  • Newtonian mechanics with constraints
  • Lagrangian mechanics with constraints
• Noether theorem:
  • Noether theorem: momentum conservation
  • Noether theorem: angular momentum and energy
• References:
  • Constraints are nicely discussed in Goldstein 2.4.
  • Noether theorem Tong 2.4.1; A good introduction to Noether is given here

Rigid body motion and accelerating frames:

• Kinematics of rotating frames
  • Kinematics of rotations (part 1)
  • Non inertial frames
  • Kinematics of rotations (part 2): Euler angles
• 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
    • Motion of a symmetric top
• 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

• Forced Oscillations:
  • Forced oscillations and secular terms
• The retarded Green function for the damped oscillator
  • Green function and transients
  • Using fourier transforms to find the response function (not covered)
  • Brownian motion
• Non linear oscillations
  • Naive perturbation theory
  • Secular perturbation theory
  • Parametric resonance
  • Nonlinear resonant oscillations
• The pondermotive force and dynamics in rapdily oscillating force field
  • Lecture and slides
  • Link to video
• Slides for non-linear oscillations: all slides
• Normal Modes:
  • The generalized e-value problem. Normal modes.
  • Example
  • Normal modes and symmetry
  • Vibrations of Molecules.
  • Movie of molecular oscillation
• 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

Hamiltonian Mechanics:

• Phase space and the Liouville theorem
• Phase space density.
• Poisson brackets
• Infinitessimal canonical transformations, and Noether's theorem revisited
• A short summary
• General canonical transformations
  • General theory
  • Symplectic Integrators
• Adiabatic invariance
  • Adiabatic invariance
  • Motion in magnetic field
  • Alternate proof
• References:
  • Starts by following Tong.
  • Then switches to Goldstein for the most part
  • Adiabatic invariance follows, Likharev and Landau.
  • Symplectic integrators follow Ruth.

Waves:

• From discrete to conintuous systems, the Euler-Lagrange equations
• Propagation of waves
• The canonical stress tensor
• Reflection of waves

The Hamilton-Jacobi Formulation:

• The onshell action
• The Hamilton-Jacobi equation
• The Kepler Problem with the Hamilton-Jacobi formulation