The Foundations Of Celestial Mechanics

The Foundations Of Celestial Mechanics

George W. Collins, II
Published in: Pachart Foundation dba Pachart
Release Year: 2004
ISBN: 978-0881-2-6009-0
Pages: 163
Edition: 1st
File Size: 2 MB
File Type: pdf
Language: English

Description of The Foundations Of Celestial Mechanics

The Foundations Of Celestial Mechanics book resulted largely from an accident. I was faced with teaching celestial mechanics at The Ohio State University during the Winter Quarter of 1988. As a result of a variety of errors, no textbook would be available to the students until very late in the quarter at the earliest. Since my approach to the subject has generally been non-traditional, a textbook would have been of marginal utility in any event, so I decided to write up what I would be teaching so that the students would have something to review beside lecture notes. This is the result.
Celestial mechanics is a course that is fast disappearing from the curricula of astronomy departments across the country. The pressure to present the new and exciting discoveries of the past quarter century has led to the demise of a number of traditional subjects. In point of fact, very few astronomers are involved in traditional celestial mechanics. Indeed, I doubt if many could determine the orbital elements of a passing comet and predict its future path based on three positional measurements without a good deal of study. This was a classical problem in celestial mechanics at the turn of this century and any astronomer worth his degree would have had little difficulty solving it. Times, as well as disciplines, change and I would be among the first to recommend the deletion from the college curriculum of the traditional course in celestial mechanics such as the one I had twenty five years ago.
There are, however, many aspects of celestial mechanics that are common to other disciplines of science. A knowledge of the mathematics of coordinate transformations will serve well any astronomer, whether observer or theoretician. The classical mechanics of Lagrange and Hamilton will prove useful to anyone who must sometime in a career analyze the dynamical motion of a planet, star, or galaxy. It can also be used to arrive at the equations of motion for objects in the solar system. The fundamental constraints on the N-body problem should be familiar to anyone who would hope to understand the dynamics of stellar systems. And perturbation theory is one of the most widely used tools in theoretical physics. The fact that it is more successful in quantum mechanics than in celestial mechanics speaks more to the relative intrinsic difficulty of the theories than to the methods. Thus celestial mechanics can be used as a vehicle to introduce students to a whole host of subjects that they should know. I feel that The Foundations Of Celestial Mechanics perhaps the appropriate role for the contemporary study of celestial  mechanics at the undergraduate level.
The Foundations Of Celestial Mechanics is not to imply that there are no interesting problems left in celestial mechanics. There still exists no satisfactory explanation for the Kirkwood Gaps of the asteroid belt. The ring system of Saturn is still far from understood. The theory of the motion of the moon may give us clues as to the origin of the moon, but the issue is still far from resolved. Unsolved problems are simply too hard for solutions to be found by any who do not devote a great deal of time and effort to them. An introductory course cannot hope to prepare students adequately to tackle these problems. In addition, many of the traditional approaches to problems were developed to minimize computation by accepting only approximate solutions. These approaches are truly fossils of interest only to those who study the development and history of science. The computational power available to the contemporary scientist enables a more straightforward, though perhaps less elegant, solution to many of the traditional problems of celestial mechanics. 
A student interested in the contemporary approach to such problems would be well advised to obtain a through grounding in the numerical solution of differential equations before approaching these problems of celestial mechanics. I have mentioned a number of areas of mathematics and physics that bear on the study of celestial mechanics and suggested that it can provide examples for the application of these techniques to practical problems. I have attempted to supply only an introduction to these subjects. The reader should not be disappointed that these subjects are not covered completely and with full rigor as this was not my intention. Hopefully, his or her appetite will be 'whetted' to learn more as each constitutes a significant course of study in and of itself. I hope that the reader will find some unity in the application of so many diverse fields of study to a single subject, for that is the nature of the study of physical science. 
In addition, I can only hope that some useful understanding relating to celestial mechanics will also be conveyed. In the unlikely event that some students will be called upon someday to determine the ephemeris of a comet or planet, I can only hope that they will at least know how to proceed.
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