Natural Sciences Elective
Order and Chaos in the Natural World
Spring Quarter 2014
April 22, 2014
I. STEWART, CHAPTER 5: ONE-WAY PENDULUM
A. Dynamics of a Pendulum Without (Much) Mathematics.
1. How well can we understand Figures 27 and 29 without making use of mathematical formulae? In other words, how well could we figure out how to draw Figures 27 and 29 without making use of mathematical formulae?
2. One method: Watch a real pendulum (possibly in the mind’s eye) and draw graphs and figures that capture the important qualitative features of the motion.
3. Another method: Think about the consequences of Newton’s second law of motion. What is the force and resulting acceleration? How does the acceleration change the velocity? How does the velocity change the position?
4. These are qualitative methods. They do not enable us to draw accurate graphs and figures. However, they do enable us to understand accurate graphs and figures.
5. Consider an ideal pendulum of the kind described by Stewart. In other words, a mass at the end of a massless, rigid rod, which is attached at the other end to a frictionless pivot. The angle a between the rod and a vertical line defines the “position” of the pendulum. (We may find it helpful to make a sketch of the pendulum and label the angle a in various situations.) By convention, the angle a is positive when the mass is displaced to our right and negative when the mass is displaced to our left.
6. If the angle a represents the position of the pendulum, how is the corresponding velocity defined or described?
B. A Standard Description of the Motion.
1. The goal here is to plot graphs of the position and velocity at different times.
2. Visualize the motion of the pendulum in the case that the amplitude of the oscillation is small. Sketch a plot of the position a of the pendulum against time.
3. Sketch a plot of the velocity of the pendulum against time. Line up the plots of the position and velocity vertically. (Hints: At what positions a does the velocity vanish? Likewise, at what positions a does the velocity have its greatest magnitude?
4. In what respects does the bottom curve in Figure 27 differ from the other curves? How might we explain those differences?
5. Now visualize the motion of a one-way pendulum. Sketch a plot of the velocity against time. Mark points on the plot where the position of the pendulum is straight down (a = 0) and straight up (a = p). Finally, sketch a plot of the position a against time.
6. Now try to visualize the motion of a pendulum in the critical case that just separates the cases of one-way motion and two-way motion. Sketch plots of the position and velocity against time.
C. Description of the Motion in Terms of Phase Portraits
1. Without referring to the formula on page 70 for the energy of a pendulum, explain the qualitative appearance of the curves in Figure 29 that represent two-way motions of the pendulum. For example, try to explain the qualitative appearance of those curves as consequences of Newton’s second law of motion.
2. Likewise, explain the qualitative appearance of the curves in Figure 29 that represent one-way motions of the pendulum.
3. Sketch the curves in Figure 29 that would represent the critical case of motions of the pendulum that just separate the cases of two-way motion and one-way motion.
4. Consider the versions of the phase portrait of a pendulum in Figures 32 and 33. Imagine that there is a mechanism that pumps energy into the pendulum. (For example, stand to the right of the pendulum and blow on it every time you see it moving to the left. Sketch or describe the trajectory of the pendulum. Is there any ambiguity about the future motion of the pendulum?
II. STEWART, CHAPTER 6: STRANGE ATTRACTORS
1. What is the essential question about the behavior of dynamical systems that is addressed in this chapter? What is the answer for a two-dimensional system (i.e., for motion in a plane)?
2. Stewart explains that Smale began with an incorrect conjecture. What is the essential content of that conjecture as portrayed by Stewart?
3. In what sense does the analysis of the van der Pol oscillator (Stewart, pp. 91-92, 137) constitute a mathematical proof that Smale’s conjecture is wrong?
4. What is the physical system represented by the solenoid constructed on pages 106-109? What are the physical laws that underlie the equations governing the behavior of that system?
III. STEWART, CHAPTER 8: RECIPE FOR CHAOS
1. What is the physical system represented by the Smale’s horseshoe constructed on pages 137-139? What are the physical laws that underlie the equations governing the behavior of that system?
2. What aspects of chaotic behavior are represented or modeled by Smale’s horseshoe?
3. We shall discuss the Henon-Heiles model, the Henon map, and the logistic map in later classes.
IV. GLEICK, pp. 34-53: REVOLUTION
1. At the beginning of this chapter, Gleick presents a brief description of Thomas Kuhn’s model of scientific revolutions. Does the history of the study of order and chaos in the solar system, as portrayed by Stewart in Chapters 2 and 4, fit the model? If so, then what parts of celestial mechanics would be identified as “normal science” in that case? And what developments in celestial mechanics would represent the revolution, and how would we describe the paradigm shift?
2. Consider the experiment with the spherical pendulum described on pages 43-44. What is it about the map representing the results of the experiment that indicates that the behavior of the pendulum is chaotic? What would be the appearance of the map if the pendulum were not chaotic?
3. According to Gleick, Stephen Smale began with an incorrect conjecture. What was the essential content of that conjecture according to Gleick?
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