This page lists what will be covered in lectures, as well as reading assignments. It also archives handouts.
To bottom (most recent lectures)| Lecture | Date | Covered in lecture | Reading (not covered or partially covered) |
| 1.1 | 9/24 |
Class organization. FW secs. 1.1-1.2: review of Newton's laws. Handout 1 (will be handed out in class). |
Read through brief recap of kinetic and potential energy:
Handout 2 . This also includes some notation that you should check that you understand. |
| 1.2 | 9/26 |
Finish angular mom. including example problem Brief recap of kinetic and potential energy using Handout 2 Solving 1-d problems using energy |
Read about reduced mass on Handout 2. |
| 2.1 | 9/29 |
2-d motion with central potential Example of $1/r$ potential (FW 1.3/1.4) Handout 3 on ellipses. |
Review derivation of Kepler's second law Complete derivation of elliptical orbit for 1/r potential |
| 2.2 | 10/1 | FW 1.5 Scattering |
Complete calculation of differential
cross section for 1/r potential to obtain quoted formula |
| 2.3 | 10/3 |
FW Ch. 2: Non-inertial frames Derivation of "fictitious" forces and simple applications |
I will cover this material quickly I suggest reading through the chapter for more background |
| 3.1 | 10/6 |
Application of fictitous forces to Foucault pendulum Is there absolute space and time? Overview and introduction to Lagrangian dynamics |
For fun: read preface and introduction to Lanczos' book (on reading list) Background for next time: read sections 13 and 14 in Ch. 3 (which I will return to later---I do not follow the order in this chapter) |
| 3.2 | 10/8 |
Calculus of variations (FW sec 17) Principle of least action for unconstrained system (FW sec 18) A trick to get a first integral (rushed) |
Read about brachistochrone problem in sec. 17 Read about the first integral in sec. 20 (it is the (minus the) Hamiltonian, which may or may not be Energy in general) |
| 3.3 | 10/10 |
Generalized coordinates and holonomic constraints (FW sec 13) D'Alembert's principle (FW sec 14) Deriving Lagrange's equation for conservative forces (eq. 15.22) |
I gave a variational derivation of (15.22) which is simpler, but less general than the derivation in FW sec 15 I worked one simple example, but you should read FW sec 16 for others |
| 4.1 | 10/13 |
Physical basis for D'Alembert's prnciple Lagrange multipliers for constrained motion Example of bead on wire |
Alternative derivation of method in FW 19 Examples in FW 19 |
| 4.2 | 10/15 |
Finish discussion of bead on wire example Meaning of Lagrange multipliers Hoop on incline example |
Read footnote on FW p.71 for details of how to proceed with differential non-holonomic constraints |
| 4.3 | 10/17 |
Brief mention of what to do with differential
non-holonomic constraints. What if non-constraint forces are not conservative? Use Lagrange's equation (FW 15) Velocity dependent potentials (EM forces) Begin small oscillations (FW ch. 4, sec.21) |
We will skip sec. 20 ("Generalized momenta and Hamiltonian") for now, returning to Hamiltonian methods later |
| 5.1 | 10/20 |
General method for small oscillations (FW sec. 22) Obtaining normal modes. |
Solve 3 spring problem for third frequency and normal mode |
| 5.2 | 10/22 |
Modal matrix and normal coordinates Example of 3 masses and 2 springs in 1 dimension. |
Read coupled pendula example (FW sec 23) |
| 5.3 | 10/24 |
Large number of degrees of freedom (FW sec 24) Taking the continuum limit (FW sec. 25) |
Solving the largeN problem using infinite matrices
(FW sec 24) Derivation of Euler-Lagrange equation in terms of Lagrange density Constructing normal coordinates explicitly (both in FW sec. 25) |
| 6.1 | 10/27 |
Begin FW Ch 5, Rigid body motion Inertia tensor, principal axes (FW 26) |
Parallel axis theorem |
| 6.2 | 10/29 |
Euler's equations (FW sec 27) Application to symmetric and asymmetric top (FW sec 28) |
Read about applications of Euler equations not discussed in class (FW sec 28) |
| 6.3 | 10/31 |
Euler angles and applications (FW secs. 30 & 31) Handout 4 . |
Read the details of nutation in FW sec. 31 (some will be needed for HW7) |
| 7.1 | 11/3 | Review session. | |
| 7.2 | 11/5 | Midterm | Solution. |
| 7.3 | 11/7 |
Discussion of midterm Hamiltonian mechanics: FW secs. 20 and 32 |
|
| 8.1 | 11/10 |
Dangers with cyclic coordinates in Lagrangian mechs. Action principle for Hamilton's equations (FW sec 32) Canonical transformations (FW sec 34) |
Example of H!=T+V and yet being constant
(FW sec 20, page 82)
Hamiltonian for charged particle (FW sec 33) |
| 8.2 | 11/12 |
Generating canonical transformations (FW sec 34) Hamilton-Jacobi equation (FW sec 35) Handout 5. |
|
| 8.3 | 11/14 |
Relation of Hamilton-Jacobi equation to QM Poisson brackets (FW sec 37) |
I will not cover Action-angle variables (FW sec 36) |
| 9.1 | 11/17 |
Introduction and overview of upcoming discussion of chaos Anharmonic oscillator and flows in phase space |
Background reading: Baker and Gollub: Chs 1 and 2, or Strogatz Ch 1. |
| 9.2 | 11/19 |
Damped pendulum---phase space flows and attractor
(using simple
Mathematica notebook ) If time, begin discussion of forced, damped pendulum ) |
Baker and Gollub: middle of Ch. 2/beginning of Ch. 3 Strogatz, sec. 6.7 |
| 9.3 | 11/21 |
Forced damped pendulum
(using
Mathematica notebook ). Limit cycles, Poincare maps, period doubling, chaos |
Baker and Gollub: first part of Ch 3 |
| 10.1 | 11/24 |
More on the forced damped pendulum
(using a new
notebook ): bifurcation diagram, period triplings, fractional windings and sensitivity to initial conditions Begin iterated maps---intro to logistic map. |
BG Ch4.1 Strogatz Ch 10.0-2. |
| 10.2 | 11/26 |
Logistic map as a model for route to chaos Self-similarity, understanding bifurctions with return functions Mathematica notebooks used in this class and next: logistic map , and return functions . |
|
| 10.3 | 11/28 | HOLIDAY | |
| 11.1 | 12/1 |
Self-similarity and universality using return maps; Lyapunov exponents; Period 3 windows |
Further reading for fun: Entropy characterization of chaos (BG 4.1.4). |
| 11.2 | 12/3 |
Definitions of fractal dimension Fractal dimension of strange attractors |
Further reading: Strogatz Ch. 11, BG Ch 5.1 |
| 11.3 | 12/5 |
Brief discussion of synchronizing metronome demo Discussion of final exam Relation of Lyapunov exponents and fractal dimension Brief comments on other examples of chaotic behavior |
For fun: Read BG Ch. 4.2 (circle map), 4.3 (Horseshoe map),
5.4 (Information change and Lyapunov exponents),
Ch. 6 (Experimental Characterization)
and 7 ("Chaos broadly applied").
These topics will not, however, be on the exam.
Also for fun, an unused notebook on the circle map and frequency locking. Other reading: Strogatz Ch. 12 |
| 12 | 12/9 | Final Exam (10:30-12:20 in A114) |