This page lists what was covered in lectures, reading assignments, and archives handouts.
To bottom (most recent lectures)| Lecture | Date | Covered in lecture | Reading (not covered or partially covered) |
| 1.1 | 9/26 |
Class organization. FW secs. 1.1-1.2: review of Newton's laws. Handout 1 . |
Finish reading through second page of 1st handout, and think about its application to the problem discussed in class Check and internalize FW eq.(2.15) Remind yourselves about Energy and conservative forces (FW p. 4 and 9) |
| 1.2 | 9/28 |
Finish angular mom. recap and example problem Brief recap of kinetic and potential energy Make sure you are familiar with the notation on second handout and are comfortable with the results. Reduced mass. |
|
| 2.1 | 10/1 |
2-d motion with central potential Example of $1/r$ potential (FW 1.3/1.4) Handout 3 (corrected) on ellipses. |
Review derivation of Kepler's second law Complete derivation of properties of elliptical orbit from integration done in class |
| 2.2 | 10/3 | FW 1.5 Scattering |
Complete calculation of differential
cross section for 1/r potential to obtain quoted formula |
| 2.3 | 10/5 | FW Ch. 2: Non-inertial frames |
I will cover this material quickly I suggest reading through the chapter for more background |
| 3.1 | 10/8 |
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 | 10/10 |
Calculus of variations (FW sec 17) Principle of least action for unconstrained system (FW sec 18) A trick to get a first integral |
Read about brachistochrone problem in sec. 17 |
| 3.3 | 10/12 |
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 give a variational derivation of (15.22) which is simpler, but less general than the derivation in FW sec 15 I do one simple example, but you should read FW sec 16 for others |
| 4.1 | 10/15 |
Deriving Euler-Lagrange equations for constrained motion Method 2: Lagrange multipliers (FW 19) Example of bead on wire |
Alternative derivation of method in FW 19 Examples in FW 19 |
| 4.2 | 10/17 |
Meaning of Lagrange multipliers (finish discussion
with example) Brief mention of what to do with non-holonomic constraints and with non-conservative constraints |
Read footnote on FW p.71 for details of how to proceed with differential non-holonomic constraints |
| 4.3 | 10/19 |
What if non-constraint forces are not conservative? Then 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 | 10/22 |
General method for small oscillations (FW sec. 22) Obtaining normal modes. |
|
| 5.2 | 10/24 |
Modal matrix and normal coordinates Large number of degrees of freedom (FW sec. 24) |
Coupled pendula example (FW sec 23) Solving the largeN problem using infinite matrices (FW sec 24) |
| 5.3 | 10/26 |
Complete discussion of large number of degrees of freedom Taking the continuum limit (FW sec. 25) |
Derivation of Euler-Lagrange equation in
terms of Lagrange density Constructing normal coordinates explicitly (both in FW sec. 25) |
| 6.1 | 10/29 |
Begin FW Ch 5, Rigid body motion FW sec 27: Inertia tensor, principal axes. |
Parallel axis theorem |
| 6.2 | 10/31 |
Euler's equations (FW sec 27) Application to symmetric and asymmetric top (FW sec 28) |
Read about applications of Euler equations < br> not discussed in class (FW sec 28) |
| 6.3 | 11/2 |
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 | 11/5 | Hamiltonian mechanics: FW secs. 20 and 32 | |
| 7.2 | 11/7 | Midterm | Solution. |
| 7.3 | 11/9 |
Discussion of midterm Dangers with cyclic coordinates in Lagrangian mechs. Action principle for Hamilton's equations (FW sec 32) Canonical transformations (FW sec 34) |
Hamiltonian for charged particle (FW sec 33) |
| 8.1 | 11/12 | HOLIDAY! | |
| 8.2 | 11/14 |
Generating canonical transformations (FW sec 34) Hamilton-Jacobi equation (FW sec 35) Handout 5. |
|
| 8.3 | 11/16 |
Relation of
Hamilton-Jacobi equation to QM Poisson brackets (FW sec 37) |
I will not cover Action-angle variables (FW sec 36) |
| 9.1 | 11/19 |
Introduction and overview of upcoming discussion of chaos Anharmonic oscillator and flows in phase space |
Baker and Gollub: Chs 1 and 2 |
| 9.2 | 11/21 |
Damped pendulum---phase space flows and attractor (using Mathematica notebook ) |
Baker and Gollub: First part of Ch. 3 |
| 9.3 | 11/23 | HOLIDAY! | |
| 10.1 | 11/26 |
Forced damped pendulum (BG Ch 3) (see Mathematica notebook ). Limit cycles, Poincare maps, period doubling, chaos |
Baker and Gollub: first part of Ch 3 |
| 10.2 | 11/28 |
Finish phenomenology of forced, damped pendulum Bifurcation diagram, period doubling route to chaos, winding number Mathematica notebooks used in class: Bifurcation plot and winding number , phase plots and Poincare sections (an extension of the notebook from last lecture). |
BG: complete Ch. 3 and the last section of Ch. 4 |
| 10.3 | 11/30 |
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 . |
BG section 4.1 |
| 11.1 | 12/3 |
More on the logistic map, using notebooks from last class:
Bifurcation and universality using return maps; Lyapunov exponents; Period 3 windows |
Not covered in class, but you should read: Entropy characterization of chaos (BG 4.1.4). There will be a HW problem on this and it is a possible topic for the final. |
| 11.2 | 12/5 |
Definitions of fractal dimension Fractal dimension of strange attractors Evaluations |
|
| 11.3 | 12/7 |
Discussion of final exam Relation of Lyapunov exponents and fractal dimension Summary of what we have learned about chaos Survey of experimental examples |
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. |
| 12 | 12/12 | Final | Solution. |