Natural Sciences Elective
Order and Chaos in the Natural World
Spring Quarter 2013
April 16, 2013
I. STEWART, CHAPTER 2: EQUATIONS FOR EVERYTHING
A. Models of the Solar System
1. Geocentric models.
b. Antikythera mechanism.
2. Heliocentric models.
b. Kepler’s laws.
3. Tychonic model.
4. What does Stewart leave out of this account of models of the solar system?
5. What is the basis on which one would prefer one model to the others?
a. Falling bodies.
b. The pendulum.
c. Galilean satellites of Jupiter; Kepler’s third law.
a. Laws of motion.
b. Law of gravitation.
C. Analytical Dynamics
d. Organ pipes.
2. Fluid dynamics.
3. Flow of heat.
4. How would mathematicians have studied such systems and determined their behaviors?
5. What is the paradigm for doing classical physics?
D. Other Issues
1. Often, solutions cannot be found exactly and in closed form.
2. Technical problems.
a. Three-body collisions.
3. Lagrangian and Hamiltonian formulations of mechanics.
4. Statistical problems.
1. The chapter is essentially a history of astronomy and physics. What are the highlights?
2. What are the attributes of classical physics that are portrayed in the chapter?
3. What limits our power to study physical systems in this way?
4. What is the role of mathematics in all of this?
5. Why is the Copernican model of the solar system preferable to the Ptolemaic model?
6. What are the issues involved in the predicting the motion of a single planet around a star? (The two-body problem.)
F. Elements of Calculus
a. Plot a curve representing a function.
b. Plot a straight line tangent to the curve at a point.
c. Define the slope of the line.
d. The derivative at the tangent point is the slope of the line.
a. Plot velocity against time.
b. Approximate the velocity curve in terms of line segments.
c. Estimate the displacement of the particle that accumulates in a given interval of time as the sum of the areas under the line segments.
d. Claim that we can improve the estimate by taking smaller intervals of time and a larger number of (smaller) line segments.
e. In the limit, the accumulated displacement is the area under the velocity curve.
3. The point is not to do the mathematics. The point is to understand the claim that is based on doing the mathematics.
G. Planetary Motion: Kepler’s Laws and Newton’s Laws
1. Kepler’s three laws of planetary motion are described briefly on page 22 of Stewart. From that description, extract a precise statement of the three laws.
2. Newton’s three laws of motion and his law of gravitation are described briefly on page 28 of Stewart. From that description, extract a precise statement of Newton’s laws of motion and law of gravitation.
3. Are Kepler’s laws or Newton’s laws the more fundamental laws of nature? Why might one set of laws be considered more fundamental than the other?
4. Are Kepler’s laws or Newton’s laws the more general laws of nature. Why might one set of laws be considered more general than the other?
5. In summary, what is the logical relationship between Newton’s laws and Kepler’s laws?
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.
II. 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. Poincare’s Methods
1. In page 59, Stewart explains that Poincare represents the state of a dynamical system in terms of a point in “some huge-dimensional phase space.” 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 “Poincare section” 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 “Hill’s reduced model” in terms of “Neptune, Pluto, and a grain of interstellar dust.” What seems to be missing in this picture?
5. What are the “footprints of chaos” that Poincare found when he investigated a surface of section for Hill’s reduced problem?
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