Vladimir Arnol’d is a master, not just of the technical realm of differential equations but of pedagogy and exposition as well. His Ordinary Differential Equations, now in its third edition, is a classic. Arnol’d writes that he has attempted to limit the ideas he presents to a bare minimum. The heart of his treatment consists of two central notions. The first is the theorem on the rectification of a vector field (from which he extracts existence, uniqueness and differentiability results); the second is the theory of one-parameter groups of linear transformations, from which springs the theory of autonomous linear systems.
Arnol’d focuses a good deal more on the applications of ordinary differential equations to mechanics than comparable texts. This is due in part to the breadth of his interest and expertise in this area but also to the sense that applications to mechanics put meat on the bare bones of the subject. The equation of the pendulum appears early and the author returns to it regularly to illustrate other concepts. So, for example, the conservation of energy appears as an example of first integrals, the small parameter method is derived as a consequence of the theorem on differentiation with respect to a parameter, and the study of the swing of the pendulum is naturally connected to the theory of differential equations with periodic coefficients.
Arnol’d also emphasizes the geometrical and qualitative aspects of the subject more than other authors. In support of this, he includes an abundance of small figures embedded in the text. He introduces phase space and phase flows early on and makes extensive use of these concepts.
The book is divided into five chapters: basic concepts (phase space, vector fields, phase flows), basic theorems (the rectification theorems and their consequences), linear systems (including a careful treatment of the matrix exponential), proofs of the main theorems, and differential equations on manifolds. The writing throughout is crisp and clear.
In this third edition, the first and second chapters have been substantially revised. There are new sections on elementary methods of integration for homogeneous and inhomogeneous first order linear equations, as well as on first order linear and quasi-linear partial differential equations. The author has also added material on Sturm’s theorems on the zeros of second order linear equations. In this new material the author once again focuses on the geometric core of the methods he describes and ties the results to applications, particularly in mechanics.
Arnol’d says that the book is based on a year-long sequence of lectures for second-year mathematics majors in Moscow. In the U.S., this material is probably most appropriate for advanced undergraduates or first-year graduate students.
An English translation (by a different translator) of the first edition of this book was originally published by the MIT Press in 1973; a revised edition appeared in 1978, and seems to be still in print. It has the same title as the book under review.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.