Mathematical Aspects of Classical and Celestial Mechanics is the third volume of the Dynamical Systems section of Springer’s Encyclopaedia of Mathematical Sciences. This English edition was prepared based on a second edition of a Russian text published in 2002. The aim is to “describe the basic principles, problems and methods” of classical mechanics with primary focus on the mathematical aspects. The authors concentrate on the “working apparatus” of classical mechanics (which seems to mean more practical methods amenable to calculation), although there isn’t a clear distinction between what is “working apparatus” and what is not.
The book is indeed encyclopedic. It doesn’t claim completeness, but it does offer the reader a view of the broad sweep of classical mechanics. Arnold, in particular, has worked in mechanics and related areas for more than forty years and is in as good as position as anyone living to create a coherent picture of this subject. The book is not a textbook, nor is it purely a reference book. There are no proofs to speak of, but if you wanted an idea of the broad scope of classical mechanics, this is a good place to visit. One advantage of the present book is that the authors are particularly skilled in balancing rigor with physical intuition.
Chapter 1 discusses the basic principles of classical mechanics, including Newtonian, Hamiltonian, Lagrangian and vakonomic mechanics and tells us how they are related. (Vakonomic mechanics, the least well-known of the four, is a model of motion in systems with constraints that was developed at least in part by Kozlov).
Chapter 2 takes up general questions of celestial mechanics including the n-body problem. The authors summarize work on regularization of collisions and the singularities of the n-body problem. They also describe Chazy’s classification of the final motions in the three-body problem. These topics have heretofore been buried fairly deeply in the technical literature, and the authors open them here to a wider audience.
Symmetries and order reduction are presented next, in a manner that emphasizes the prominent role of Noether’s theorem — the relationship between symmetries and conserved quantities — in modern classical mechanics. Following that, the authors discuss variational principles in the context of classical mechanics. Among other things, these variational methods provide tools to establish existence of non-trivial periodic orbits.
Integrable systems in classical mechanics are roughly those for which the equations of motion can be solved by quadrature. Although many interesting systems are non-integrable (the n-body problem for n > 2, for example), integrable systems and perturbations of integrable systems are found in many applications and are the subject of much research. Much of the latter half of the book is devoted to questions of integrability. This includes a treatment of KAM (Kolmogorov-Arnold-Moser) theory that provides converging methods for integrating perturbed Hamiltonian systems. A short section at the end discusses the interesting question of topological and geometrical obstructions to integrability.
A chapter on small oscillations in mechanical systems provides a brief but fairly comprehensive discussion of reduction to normal forms near equilibria or closed trajectories. A concluding section aimed more at specialists focuses on tensor fields in phase space invariant under the equations of motion.
The authors provide an extensive bibliography and a well-selected set of recommended readings. Overall, this is a thoroughly professional offering.
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.