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The Geometry of Celestial Mechanics

Hansjörg Geiges
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Student Texts 83
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Anil Venkatesh
, on

The Geometry of Celestial Mechanics offers a fresh look at one of the most celebrated topics of mathematics. In this text, the author attempts to introduce the topic in an elementary setting compared to Siegel and Moser’s Lectures on Celestial Mechanics, which he describes as taking a more mathematically mature approach. Nonetheless, this text contains more than enough content for a semester’s course, as detailed below.

The book is nominally intended for an audience of “second-year mathematics or physics students all the way up to Ph.D. students.” This surprisingly broad range is perhaps due in part to the fact that most students are unfamiliar with the basic terminology of celestial mechanics. While advanced students will be well versed in the geometric content of the first few chapters, the historical context and application domain are sufficiently novel to keep such students engaged.

On the whole, the book succeeds in covering the main topics of celestial mechanics while still assuming very little background in geometry. For example, chapter 2 is a succinct and attractive introduction to conic sections, leading into chapter 3’s treatment of the Kepler problem. Likewise, chapter 8 introduces such concepts as stereographic projection, inversion, and hyperbolic and projective geometry in order to support the treatment of Hamiltonian mechanics and the topology of the Kepler problem in chapters 9 and 10. One of the strongest aspects of the exposition is the author’s commitment to a holistic treatment of the Kepler problem. Several geometric proofs of the first law are given, and Newton’s solution of the Kepler equation using the curtate cycloid is provided. The case of parabolic trajectories is solved explicitly by means of a surprising but engaging detour into the solution of cubics by radicals. On the whole, the text achieves self-contained content coverage without becoming ponderous in the details.

Every chapter concludes with a suite of instructive exercises, supporting the author’s claim that the book is suitable for self-study. Many of the exercises are computational in nature, which should greatly aid students who are less experienced in geometry. All chapters but the second also contain a page or more of historical notes and references. This practice contributes to my favorite aspect of the text: it is eminently readable. I was surprised to find myself leafing through the book as I would a novel; in fact, I intend to give this book as a “light reading” gift to several colleagues.

My only reservation about the book relates to the author’s claim that it can be covered in a 14-week course. While this would certainly hold for a graduate student audience, I am uncertain whether a typical undergraduate would be able to master the content at this speed, particularly the more advanced topics in chapters 9 and 10. This critique may especially hold at American-style universities where students often take several semesters of engineering calculus before being exposed to rigorous analysis.

While the book is indeed self-contained in its development of geometry, an undergraduate to whom these concepts are entirely new might need longer than anticipated to internalize to content, a step that is crucial to understanding the main results of the text. For this reason, I find the book in its entirety to be better suited to an advanced undergraduate topics course than as an introduction to geometry for second-year students and up. On the other hand, excluding some of the more advanced content would result in an attractive introduction to geometry for students who are newer to the discipline. I expect that the introduction to hyperbolic and projective geometry in the context of celestial mechanics would be particularly well received.

I would gladly recommend this book for two different uses. If you are interested in celestial mechanics but have never learned the details, this book is a quick and highly enjoyable read. From a student perspective, this book is suitable for advanced undergraduates, or even as an introduction to geometry if some topics are omitted.

Anil Venkatesh is an assistant professor of mathematics at Ferris State University.

1. The central force problem
2. Conic sections
3. The Kepler problem
4. The dynamics of the Kepler problem
5. The two-body problem
6. The n-body problem
7. The three-body problem
8. The differential geometry of the Kepler problem
9. Hamiltonian mechanics
10. The topology of the Kepler problem