MLA 31500

Natural Sciences Elective

 

Order and Chaos in the Natural World

 

Spring Quarter 2008

 

CLASS NOTES

SECOND CLASS

April 5, 2008

 

In thinking about the questions below, consider how you would make use of the principles described in Stewart in order to formulate strategies or procedures for constructing answers.  Although the answers are important, it is the process of finding the answers that is really interesting.

 

I.     STEWART, CHAPTER 4: THE LAST UNIVERSALIST

 

A.   Equilibrium and Stability

 

1.    Consider a bowl of the form of a hemisphere and a ball bearing free to roll about in the bowel.  Describe a situation in which the bearing is in static equilibrium (i.e., at rest).  Is this a stable state of equilibrium?  How can you tell?

 

2.    Now turn the bowl upside down.  Again the ball bearing is free to roll about on the inverted bowl.  Is there a static equilibrium state in which the bearing is at rest?  Describe it.  Is this a stable state of equilibrium?  How can you tell?

 

3.    Now turn the bowl upright.  If there were no friction or air resistance, then we could get the bearing to roll at a constant rate around the bowl on a horizontal circular trajectory.   This would be a state of dynamical equilibrium in which the force of gravity, the centrifugal force, and the force exerted by the bowl just balance to keep the bearing in a steady motion.  How would one test this state for stability?  If the system were stable, then what would happen?  What would happen if it were not stable?

 

B.   Stability of the Solar System

 

1.    Imagine that we could investigate the stability of the solar system experimentally with the aid of a time machine.  In order to perform the experiment, we visit the solar system two billion years in the future.  Describe the arrangement and motions of the planets that you would expect to observe if the solar system were stable.

 

2.    Describe the arrangement and motions of the planets that you might expect to observe if the solar system were unstable.

 

3.    Is this concept of the stability of the solar system the same as the concept considered above of the stability of states of the ball bearing free to roll on the surface of the bowl?

 

C.   PoincareÕs Methods

 

1.    In page 59, Stewart explains that Poincare represents the state of a dynamical system in terms of a point in Òsome huge-dimensional phase space.Ó  Moreover, the motion of the system is represented by a curve traced out by that point in the phase space.  Consider a single planet moving in the gravitational field of the sun.  Describe the phase space in which Poincare would represent the motion of the planet.  How many dimensions would that phase space have?  What are those dimensions?

 

2.    What are the principles or laws that determine the curve in the phase space representing the motion of the system considered in the preceding section?

 

3.    Stewart then describes the use of a ÒPoincare sectionÓ in order to find periodic orbits of the system.  For the case considered above of a single planet, describe a possible Poincare section.  Suppose we could watch the point representing the state of the system move about in the phase space.  How might we use the Poincare section in order decide whether or not the motion is a periodic orbit.

 

4.    Stewart describes ÒHillÕs reduced modelÓ in terms of ÒNeptune, Pluto, and a grain of interstellar dust.Ó  What seems to be missing in this picture?

 

5.    What are the Òfootprints of chaosÓ that Poincare found when he investigated a surface of section for HillÕs reduced problem?

 

II.   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 mindÕs eye) and draw graphs and figures that capture the important qualitative features of the motion.

 

3.    Another method:  Think about the consequences of NewtonÕs 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.  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 ÒpositionÓ 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 NewtonÕs 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?

 

 

LINKS:

 

Return to Course Page: mla315spring2008.htm

 

Return to Peter Vandervoort's Home Page: pov.html

 

 

Go to the home page of the Department of Astronomy and Astrophysics

of the University of Chicago:  http://astro.uchicago.edu/