The N-body problem — how to determine the motion of N point masses interacting according to mutual Newtonian gravitational attraction - is a mathematical mountain. It has served as a goad to mathematical development over the course of two centuries. It has also provided a test bed for new mathematical ideas, particularly since the time of Poincaré. While the N-body problem could the basis of a sizable volume all by itself, the current book takes a different path. The primary subject here is the basic theory of Hamiltonian differential equations studied from the perspective of differential dynamical systems. The N-body problem is used as the primary example of a Hamiltonian system, a touchstone for the theory as the authors develop it.
This book is intended to support a first course at the graduate level for mathematics and engineering students. It assumes a decent undergraduate background in differential equations, linear algebra and advanced calculus. It is a well-organized and accessible introduction to the subject, easier to get into (and more narrowly focused) than alternatives such as Abraham and Marsden’s Foundations of Mechanics and better suited to classroom use. The problem sets are interesting and well-chosen; problems are challenging but not discouragingly difficult.
Of the first eight chapters, all but the fourth and seventh compose the core of the basic material. The book begins with an introduction to Hamiltonian dynamics via harmonic oscillators and other simple systems. The second chapter focuses on the equations of celestial mechanics and includes a discussion of the general N-body problem (including, for example, the classical integrals-of-the-motion, central configurations, and Lagrangian solutions) as well as the restricted 3-body problem and Kepler’s problem. The next two chapters concentrate on the linear theory of Hamiltonian systems; the first addresses basic material and the second discusses more specialized topics. Chapters 5 and 6 take up exterior algebra, differential forms and symplectic transformations, but the treatment is pretty minimal — just enough to hit the high points. Here, as elsewhere, the authors refer the reader to other texts for more detail and sometimes for proofs.
Chapter 7 is more specialized; it deals with special coordinates and would be of most value to someone planning continued work in this field. The geometric theory — a self-contained introduction to dynamical systems — follows, and completes the core material. The remaining six chapters concern mostly special topics. These include continuation of solutions, normal forms, bifurcations of solutions, variational techniques and elements of KAM (Kolmogorov-Arnold-Moser) stability theory.This is an attractive book, especially locally. Globally, the reader might wish for a better thread to follow when moving from chapter to chapter. A lot of the “why” questions get short shrift. One might argue that a graduate student should supply his or her own answers to the why questions, but a specialist in a field is in a unique position to offer wise counsel (i.e., valuable insight and perspective). A paragraph here and there that ties the threads of the narrative to provide a more global perspective could make a real difference.
Hamiltonian Differential Equations and the N-Body Problem.- Exterior Algebra and Differential Forms.- Symplectic transformations and coordinates.- Introduction to the Geometric Theory of Hamiltonian Dynamical Systems.- Continuation of Periodic Solutions.- Perturbation Theory and Normal Forms.- Bifurcations of Periodic Orbits.- Stability and KAM Theory.- Twist Maps and Invariant Curves.