MLAP 31500

Natural Sciences Elective

Order and Chaos in the Natural World

Spring Quarter 2011

Writing Assignment I

Due May 3, 2011

The first writing assignment of the course is to write an abstract of a chapter in Does God Play Dice? other than Chapter 1 or 2. The abstract should consist of a single paragraph which describes with extreme brevity the main points of the chapter. These might include, as appropriate, such matters as the topics to be covered, the organizing principles that underlie the choice of topics and their arrangement, the principal claims and arguments, and the relation of the chapter to be abstracted to other chapters in Does God Play Dice? Make use of technical nomenclature consistently with the usage in the text. Assume that nomenclature and general concepts in science are known to the reader, and omit explanations and definitions of such terms and ideas except where they might be essential to an emphasis of particularly important points in the chapter. In general, an abstract of 500 words or fewer would be appropriate, and every effort should be made to keep it under 700. An abstract of Chapter 2 is provided separately as a model; it amounts to a few more than 400 words. It also illustrates the format that will be appropriate for abstracts.

Abstracts will be evaluated with respect to the following five equally weighted categories.

TOPICS: The abstract should identify the main topics of the chapter.

ORGANIZING PRINCIPLES: The abstract should identify or describe the principles that govern the choice of topics, their arrangement, etc.

CLAIMS AND ARGUMENTS: The abstract should describe the author’s claims regarding the historical significance and scientific role of the developments described in the chapter, and it should describe briefly the author’s reasons for asserting those claims.

RELATION TO OTHER CHAPTERS: The abstract should place the chapter in the larger context of the book.

NOMENCLATURE: The use of technical terms and references to scientific principles should be consistent with the technical and scientific usage adopted by the author. Thus definitions are ordinarily not required, and references to scientific principles known to the appropriate reader should not include statements of those principles.

It should be emphasized that an abstract is a brief description of the contents of a text. It is neither a summary of that text, a review of the text, nor a commentary on the text. In particular, the abstract should not contain personal reflections or judgments that are not in the text.

Ian Stewart, Does God Play Dice? The New Mathematics of Chaos (Malden MA: Blackwell Publishing).

Abstract of Chapter 2, Equations for Everything

This chapter contains an historical account of developments in astronomy, mathematics, and physics that led to our modern understanding of the deterministic behavior of physical systems. A search for order and regularity in natural phenomena is an organizing principle underlying that history. The cyclic character of astronomical phenomena was recognized by ancient Greek scholars and incorporated into the Ptolemy’s geocentric model of the solar system and the Antikythera mechanism, a remarkable mechanical device for the calculation of celestial motions. The Ptolemaic model of the solar system prevailed until the Renaissance, when Copernicus formulated the heliocentric model. Kepler refined the Copernican model with the discovery of his three laws of planetary motion. Kepler’s laws provided a precise quantitative description of planetary motion, but they lacked an underlying physical theory. At the same time, Galileo developed important principles of kinematics and dynamics in his studies of the motions of pendulums and falling bodies. The unification of theoretical studies of celestial motions and terrestrial dynamics was accomplished by Newton in terms of his three laws of motion and his universal law of gravitation. The invention of the calculus by Newton and Leibniz provide the mathematical tools required for the extensive investigations of physical problems that took place during the eighteenth and nineteenth centuries. A typical investigation consisted of constructing the differential equations that represent the physical laws governing the behavior of the system under study and then searching for solutions of those equations. A theory of vibrating strings was developed in that way by Taylor and d’Alembert, and investigations of the vibrations of drumheads and bells by Euler and of organ pipes by Bernoulli soon followed. Lagrange began to develop a comprehensive theory of acoustics. The theory of heat flow developed by Fourier and investigations of the equations of gravitation by Laplace and Poisson produced fundamental new developments in mathematics. The uniqueness of the solutions of the differential equations implied that the evolutions of the systems described by those equations were deterministic. In the cases in which the equations could be solved, the solutions represented the behaviors of systems as regular. In many cases, however, solutions of the differential equations could not be found. Nevertheless, regular behavior of the kind exhibited by soluble systems came to be regarded as the normal behavior of most physical systems. The chapter includes brief accounts of reformulations of Newtonian mechanics by Lagrange and Hamilton in terms of generalized coordinates and generalized momenta.

Peter O. Vandervoort

March 31, 2011

LINKS:

Return to Course Page: mla315spring2011

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/