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Notes
Current
Notes, Fall, 2014
Lagrangian mechanics
The
2d oscillator and central forces
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).
Central forces
Central
forces I: Center of mass coordinates and conserved quantities
Central
forces II: Gravitation
Hamiltonian Mechanics
Phase
space and Hamilton's equations
Conservation
laws and the symplectic form
Connections
with quantum mechanics
Notes on
Goldstein, Fall 2012
General
coordinates and constraints
Properties
of the Levi-Civita tensor
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:
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)
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