BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

The photographs of Recife which are scattered throughout this book reveal an eclectic mix of old colonial buildings and sleek, modern towers. A clear hot sun shines alike on sixteenth century churches and glistening yachts riding the tides of the harbor. The city of 1.5 million inhabitants is the capital of the Pernambuco state in northeastern Brazil and home of the Federal University of Pernambuco, where the lectures which comprise the body of the book were delivered. Each lecturer presented a focused mini-course on some aspect of contemporary classical mechanics research at a level accessible to graduate students and later provided a written version for the book. The lectures are as varied as their authors. Taken together they constitute an album of snapshots of an old but beautiful subject. Perhaps the mathematical study of mechanics should also be dated to the sixteenth century, when Galileo discovered the principle of inertia and the laws governing the motion of falling bodies. It took the genius of Newton to provide a mathematical formulation of general principles of mechanics valid for systems as diverse as spinning tops, tidal waves and planets. Subsequently, the attempt to work out the consequences of these principles in specific examples served as a catalyst for the development of the modern theory of differential equations and dynamical systems. Part of the tradition of the subject is that the examples themselves are given center stage. Each mechanical system has special features which give it its unique character. Such features are often exceedingly interesting and beautiful, but can easily be missed if one approaches the system as a mere special case of a general theory. The Recife lectures reflect this spirit. For example, Alain Albouy provides a fascinating account of the classical twobody problem of celestial mechanics. The problem could be treated as a simple case of reduction of a Hamiltonian system with symmetry [5], [7]. For motion in the plane, the relative position of the bodies is described by a vector x ∈ R \ 0, so it is a system of two degrees of freedom. Taking into account the corresponding velocity variables (or rather, from the Hamiltonian viewpoint, the momenta), one finds that the phase space is the cotangent bundle T ∗(R2 \ 0), a four-dimensional symplectic manifold. The rotation group SO(2) acts as a symmetry group, and one expects by general theory that one can use this symmetry to reduce to a system of one degree of freedom. Using the conservation of energy, it is possible to explicitly solve such a reduced system and then to recover the solutions on the original four-dimensional phase space. In a typical problem of this kind, the result is a foliation into twodimensional invariant tori which support quasi-periodic motions. But this general analysis makes no use of the special Newtonian force law; it is equally valid for any two-body interaction with SO(2) symmetry. For the Newtonian 1/r force law, a miracle occurs — all of the solutions are periodic instead of just quasi-periodic. To put it another way, the two-dimensional tori are further decomposed into invariant circles. This highly degenerate situation seems unbelievable from the point of view