MLA 31500

Natural Sciences Elective

 

Order and Chaos in the Natural World

 

Spring Quarter 2008

 

CLASS NOTES

FIRST CLASS

March 29, 2008

 

 

I. INTRODUCTION

 

A.     Housekeeping: Handout

 

1.      Course.

 

2.      Instructor.

 

3.      Texts.

 

4.      Organization.

 

(a)     Discussion.

 

(b)    Lecture demonstrations.

 

(c)     Papers.

 

(d)    No exams.

 

5.      Meeting times.

 

6.      Attendance.

 

7.      Liberal learning.

 

B.      Introductions

 

1.      Student Data forms.

 

2.      Introduce ourselves.

 

C.      Class Notes distributed in class and posted on the course web page <http://astro.uchicago.edu/~voort/pov.html> are essentially agendas for class discussions.  They will consist mainly of brief outlines of subject matter or questions about the subject matter.  The questions are intended to serve as guides in reading assigned material and as aids in preparing for class discussions.  Therefore, it is much more important to think about how one might arrive at the answers to the questions than it is to work out the answers explicitly.

 

 

II.   DETERMINISTIC SYSTEMS, RANDOM PROCESSES, AND CHAOS ACCORDING TO IAN STEWART AND DAVID RUELLE

 

A.   General Considerations

 

1.    What do our Stewart and Ruelle accomplish in the introductory chapters of their books?

 

2.    What should we understand as the meanings of the terms Òdeterministic systems,Ó Òrandom processes,Ó and Òchaos?Ó  Do Stewart and Ruelle agree on the meanings of these terms?

 

3.    What might be examples of a deterministic system, a random process, and chaos?

 

4.    The book Chaos by James Gleick has the subtitle Making a New Science.  Is the study of order and chaos a science?  Or is the study of order and chaos a part of a particular scientific discipline such as astronomy, biology, mathematics, physics, or another discipline?

 

B.   Deterministic Systems in the Natural World

 

1.    What is the function of mathematics in the study of the natural world?

 

2.    What does the discovery of Òlaws of natureÓ contribute to an understanding of the natural world?

 

3.    In what ways are mathematical formulations of the laws of nature useful?  In particular, for what purposes do we reduce the laws of nature to differential equations?  (By the way, what is a differential equation?  As a student of liberal learning, try to answer this question without making use of technical aspects of mathematics.)

 

4.    What is the connection between the reduction of the laws of nature to differential equations and the conclusion that those laws describe deterministic systems?

C.   The Encounter of Voyager 1 and 2 with Hyperion.

 

1.    Give a precise description of the hypothetical experiment presented in the section ÒVoyage to HyperionÓ in Does God Play Dice?

 

2.    What is the expected result of that experiment?  Why?

 

3.    What is the ÒobservedÓ result?

 

4.    What is the apparent paradox revealed by a comparison of the expected and observed results?

 

5.    What does this discussion suggest about the relationship between the determinism of a physical system and the predictability of that system?

 

D.   Introducing Chaos

 

1.    Stewart illustrates his introductions to chaos by describing the behaviors of a particular dynamical system or model of a dynamical system.  What is that illustrative example?  Why does that model deserve to be called a dynamical system?

 

2.    In what respects is the system that Stewart uses in order to illustrate chaotic behavior similar to Hyperion?  In what respects are those systems different?  Which is the more realistic system with which to illustrate chaotic behavior?  Why?

 

3.    In what respects are systems governed by discrete mappings and systems governed by differential equations similar?  In what respects are they different?

 

4.    Ruelle reaches the point of introducing the subject of chaos by much lengthier route.  Manifestations of chaos first appear in Chapter 7, and an explicit discussion of chaos does not appear until Chapter 11.  Why is it useful, nevertheless, to begin reading Ruelle now?

 

 

I.     STEWART, CHAPTER 2: EQUATIONS FOR EVERYTHING

 

A.   Models of the Solar System

 

1.    Geocentric models.

 

a.     Ptolemy.

 

b.    Antikythera mechanism.

 

2.    Heliocentric models.

 

a.     Copernicus

 

b.    KeplerÕs 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?

 

B.   Dynamics

 

1.    Galileo.

 

a.     Falling bodies.

 

b.    The pendulum.

 

c.     Galilean satellites of Jupiter; KeplerÕs third law.

 

2.    Newton.

 

a.     Laws of motion.

 

b.    Law of gravitation.

 

c.     Calculus.

 

(i)   Differentiation.

 

(ii)  Integration.

 

C.   Analytical Dynamics

 

1.    Vibrations.

 

a.     Strings.

 

b.    Bells.

 

c.     Drums.

 

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.

 

b.    Singularities.

 

3.    Lagrangian and Hamiltonian formulations of mechanics.

 

4.    Statistical problems.

 

E.    Questions.

 

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

 

1.    Differentiation.

 

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.

 

2.    Integration

 

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.

 

 

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/