PHYS 252

Always hand in:
  1. written solutions to any questions
  2. a paper print-out of well-commented code. Include a multiline comment at the top of your code with (i) the assignment name, (ii) your name, and (iii) the date you handed in all elements of the assignment
  3. paper print-out of output (graph or text)
  4. also, e-mail me (y-lithwick@northwestern.edu) the code with your name and the exercise number in the subject line

Assignment #8
[10 pts, due 2pm, April 26]

  1. Reproduce figures 3.6 (left panel) and 3.8 (both panels) in the text. For Fig 3.6, approximately how high does FD have to be to give chaos? Show plots with FD above and below that critical value. Choose a parameter (e.g., q or FD), and explore what happens when you change it. Show me a few plots and explain briefly what is going on.
  2. Reproduce figure 3.7 (both panels) in the text. Make a few more such plots with different initial conditions (but the same parameters). Estimate the Lyapunov times for the two cases, showing how you arrive at your answers.
  3. Make a movie of the simulation used for the right panel of figure 3.7. The movie should show both pendula simultaneously, on the same pivot. To receive credit for this question, show me the movie at the beginning of class on the day this assignment is due (or earlier).
  4. Poincaré sections: reproduce figure 3.9 in the text. Also make a Poincaré section of the trajectory in the left-hand panel of Figure 3.8, after the initial transient has died away. Explain in both cases what the sections mean about the motion of the pendulum. How does the Poincaré section change if you change the initial condition?
  5. You need only do one of the following. (Though I encourage you to do more, and/or follow your own explorations or the book's Section 3.4.) You will be presenting your results to the class on Thursday. Find some way to get a copy of your plots or movie to the projector, e.g., via the computer in the lab that accesses the projector directly.
    1. Frank & Xiaowen: What happens to the Poincaré section of question 4 for different values of FD? Focus especially on what happens near transitions from regular to chaotic behavior. Can you find multiple transitions?
    2. Thomas & Max: Explore the Poincaré section of question 4 further by examining plots for different values q.
    3. Adam: Run simulations so that you can zoom into portions of Fig. 3.9 and show what happens at very high resolution. You should see the lines breaking up into thinner lines. You will have to run the code for a long time to get good enough resolution.
    4. [If anyone prefers they may do this one instead; it is a lot easier than it sounds:] Make a movie of the double pendulum. The equations of motion are here.