Course Information

Syllabus

 

Bibliography

Homework

 

Notes

 

 

Notes

 

Current Notes, Fall, 2014

 

Lagrangian mechanics

 

Review of Newtonian mechanics

 

The 2d oscillator and central forces

 

Beyond the Second Law

 

The action functional

The Notes, Beyond the Second Law, above present five reasons for considering alternative formulations of NewtonŐs second law. We now examine each within the context of Lagrangian mechanics.

 1. General coordinate invariance: Coordinate covariance of the Euler-Lagrange equation

2. Symmetry and the resulting conservation laws:

NoetherŐs theorem and conservation of momentum

NoetherŐs Theorem continued: conservation of angular momentum and energy, consequences of scale invariance

3. Constraints: Constraints and Lagrange multipliers

4. There exist problems that resist solution using the second law: The Brachistochrone

5. The connection with quantum mechanics will be established when we discuss Hamilton-Jacobi theory

Some worked examples: AtwoodŐs machine and the raindrop problem

 

Rotations and angular momentum

Notational conventions: index notation, covariant and contravariant vectors, and the Einstein convention

The special orthogonal group,  SO(3).

The special unitary group, SU(2).

Study Guide for Midterm

 

 

Central forces

 

Central forces I: Center of mass coordinates and conserved quantities

 

Central forces II: Gravitation

 

Perihelion advance

 

Yukawa potential

 

 

Hamiltonian Mechanics

 

Phase space and Hamilton's equations

 

Conservation laws and the symplectic form

 

Canonical transformations

 

Hamilton-Jacobi theory

 

Connections with quantum mechanics

 

 

 

 

 

Notes on Goldstein, Fall 2012

 

Review of Newtonian mechanics

 

General coordinates and constraints

 

Central Forces

 

BertrandŐs Theorem

 

Brachistochrone

 

Scattering

 

Rigid Bodies: Rotations

 

Dynamics of Rigid Bodies

 

Foucault Pendulum

 

Properties of the Levi-Civita tensor

 

Hamiltonian Mechanics

 

Hamilton-Jacobi Examples

 

Additional materials

 

The use of differential forms is not required for this class, but if you are interested in pursuing them further, here is an excerpt on the subject. A little bit of more advanced material is assumed, but much of this is accessible to you:

 

Differential Forms

 

Mechanics Book

 

I have written the following text for an advanced mechanics course. Treat it as a work in progress, and please let me know of any errors you find. You will find parts of it a nice supplement to Goldstein.

 

Not-so-classical Mechanics  (Wheeler, 2005)

 

Course Information

Syllabus

 

Bibliography

Homework

 

Notes