PHY4222 Mechanics 2 - Spring 2014

Links to useful sites

1. Noether's theorem

Short article by John Baez

Long article in Wikipedia

2. Biographies

Hamilton

Lagrange

Noether

3. The tide prediction for D-Day

Physics Today article by Bruce Parker

4. Foucault's pendulum

Wikipedia article with an animation

YouTube video (Foucault's pendulum in the Houston Museum of Natural Science)

5. Free precession applet

In English (used in class)

In Russian

6. Three body problem in gravitation

Eugene Butikov's applet (applet 1 contains chaotic examples shown in class)

Same in Russian

David Harrison's applet (shown in class)

Joachim Koeppen's applet

7. Driven damped pendulum applets

MyPhysicsLab applet

In this applet, use the default values m=1, L=1, and g=1. Then omega0 will be 1, and the "damping" will be Taylor's 2beta. Since omega0=1, Taylor's choice beta=omega0/4 will correspond here to "damping"=0.5, the default value. Also, since omega0=1, the "drive frequency" will be Taylor's omega/omega0. His choice omega0=1.5*omega will therefore correspond to "drive frequency"=1/1.5=0.6666..., also the default value. omega0=1 also makes the "drive amplitude" here the same as Taylor's dimensionless parameter gamma.

Examine the angle phi as the function of time and also the angular velocity phi-dot as a function of angle phi for the following values of the drive amplitude:

0.9 (see Taylor's Fig. 12.3 on page 466)

1.06 (see Fig. 12.4 on page 467)

1.073 (see Fig. 12.5 on page 471 for gamma=1.07)

1.077 (see Figs. 12.7 and 12.8 on page 471) You will need patience plus many trials to see the period 3 behavior observed by Taylor.

1.081 (see Fig. 12.8 on page 472)

1.0826 (see the same figure) In the applet, you will probably have to look at the phi-dot vs phi graph.

1.105 (see Fig. 12.10 on page 476)

1.13 (see Fig. 12.24 on page 481)

1.15

1.35

1.4 (see Figs. 12.25 and 12.26 on pages 492 and 493)

1.45

1.503 (see Figs. 12.15 and 12.16 on pages 482 and 483)

Nasser Abbasi's program for CDF Player

Download this program, which shows Poincare section and bifurcation map, in addition to a phase-space plot (phi-dot vs phi) and a time series. Note that the bifurcation map here is phi-dot vs drive amplitude, not phi vs drive amplitude. Similarly, the time series is phi-dot vs time, not phi vs time.

The program requires CDF Player installed on your computer. It can be downloaded along with Abbasi's program.

8. PHY 4803L chaos webpage

Click here.

Look at the animated Poincare sections for the driven-damped-pendulum experiment in the PHY 4803L lab.

9. Logistic map

The applet used in class can be download from here.

For the history of the logistic map, go to the History of Mathematics website and click the link to the pdf file.

10. Websites of chaos and nonlinear-dynamics research groups(a random selection)

University of Maryland

University of Texas

UC Santa Barbara

Caltech

Last updated: 04/17/2013