## Thursday, January 13, 2011

### Blast from the past

For "Mechanics of Particles" (basically graduate mechanics and dynamics), I'd said:
- Brief review of variational calculus
- Lagrangians and Hamiltonians, action principle
- Canonical transformations, phase space
- Symmetries and conservation laws (Noether's thm?)
- Normal modes, harmonic oscillator review
- Rigid body motion (numerical work?)
- Orbital mechanics review
- Classical perturbation theory (w/ orbits, rigid body dynamics, anharmonic oscillator)
- Action-angle variables
- Poisson brackets, symplectic structure (*definitions of 1-forms, tangent spaces, tangent bundles?)
- Chaos, nonlinear dynamics, ergodicity
- Brief review of Einstein summation convention
- Special relativity w/ Einstein summation convention, space-time diagrams

For "Continuum mechanics" (fairly unique, I now realize - many departments offer no such course), my suggestions reflected my undergrad engineering background to some degree. I now realize that what I list below is considerably too much for a 10 week course:
- Mechanics of solids:
+ Continuum mechanics version of Hooke's law; stress, strain, tension, compression, shear, bulk modulus, a few numbers about strength of materials, Young's modulus, shear modulus
+ Lagrangian/Hamiltonian densities, more variational calculus
+ *Flexure of beams, bending moments, areal moments of inertia (why I-beams are stiffer than rods of the same cross-sectional area)
+ *Torsion of members, polar "moments of inertia"
+ *Dynamics of beams: the wave equation, longitudinal and transverse sound, natural frequencies of cantilevers
+ Acoustics, idea of acoustic impedance and mismatch
- Fluid statics
+ Hydrostatics, Archimedes' principle, buoyancy
+ *Surface tension, capillary action, wetting
- Fluid mechanics
+ Euler and Lagrange pictures
+ "Convective derivatives", transport of momentum and energy
+ The energy equation, the momentum equation, the continuity equation, the Navier-Stokes equation
+ Inviscid, incompressible flow:
- Bernoulli's Eqn.
- Potential theory
- *Vorticity, circulation, Magnus' law, "lift"
+ Viscous, incompressible flow:
- Definition of viscosity, comparison w/ shear modulus, definition of Newtonian fluid
- Stoke's law
- Intro to dimensional analysis, Reynolds' number
- Laminar flow, parabolic velocity profile in a round pipe
- Turbulent flow, mention engineering approach to these problems (Moody chart, friction factor, Bernoulli w/ losses)
- Froud number, hydraulic jumps (example of a "shock" discontinuity that you can demonstrate in a sink)
+ Compressible flow
- Mention of shockwaves, scaling

For "Statistical Mechanics", the main challenge was dealing with the divergent backgrounds of incoming students - some people had very strong undergrad preparation in statistical and thermal physics, others much less so. This is an issue in graduate quantum mechanics to an even greater degree. Now that I've taught undergrad stat mech several times, I think what I listed below could use some additional advanced topics:
- Definition of entropy, why it's a log
- The equal prob. postulate/ergodic thm.
- The Boltzmann factor and the partition fn., Fermi and Dirac distributions
- *Mention of Feynman diagram methods, saddle-point integration to get Z in complicated systems
- The canonical and grand canonical ensembles, the chemical potential
- "Natural" variables, Legendre transforms, thermodynamic potentials, *the idea of a constrained maximization of S, the Maxwell relations, the "thermodynamic square"
- Gases
+ Ideal classical
+ Van der Waals, virial coefficients
+ Fermi gas at zero and finite T
+ Ideal Bose gas, BEC, phonons & photons *(incl. laser discussion!)
- Liquids - diagrammatic methods of treating interactions?
- Solids
+ Phonons
+ Concept of long-range order
- *Correlation functions, *connection w/ susceptibilities
- *Correlations and fluctuations, *how they're measured!
- Theories of phase transitions
+ Concept of order parameter
+ Ginsberg-Landau theory, diff. betw. 1st and 2nd order, extensions to include fluctuations
+ 1st order: Van der Waals reprise, Clausius-Clapeyron
+ Mean-field theory, example of magnetism
+ Ising model in 1-d
+ Renormalization group to solve Ising model, critical behavior, correlation length ideas
- *Transport
+ *Boltzmann equation
+ *Noise in transport: fluctuation/dissipation thm
It was definitely interesting to me to see how my thinking on this stuff has evolved now that I have to teach it. 