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9R7. Classical Many-Body Problems Amenable to Exact Treatments. (Solvable and/or Integrable and/or Linearizable…) in One-, Two-, and Three-Dimensional Space. Lecture Notes in Physics, Vol m66. - F Calogero (Dept of Phys, Univ of Rome “La Sapienza,” p Aldo Moro, Rome, 00185, Italy). Springer-Verlag, Berlin. 2001. 749 pp. ISBN 3-540-41764-8. $79.95.

Reviewed by M Pascal (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Paris, 75252 Cedex 05, France).

This book is concerned with integrable problems in classical mechanics (excluding quantam or relativistic mechanics). These problems are related to the motion of a system of particles acted upon by several kinds of forces in one-, two-, or three-dimensions. More general Hamiltonian systems (not associated with a Lagrangian function) are also considered. In most cases, integrable Hamiltonian systems mean Liouville integrable systems. In almost all examples dealing with the motion of a set of particles, the masses are the same for all particles. Several methods to find integrable problems are described, but the main part of the book is devoted to the Lax pair technique. This book provides a rather exhaustive survey of results obtained in the past by scientists such as Moser, Toda, Flashka, and also by the author himself.

The book involves five chapters followed by several appendices and a rather wide list of references. Each chapter includes a great amount of examples and exercises. Due to the rather abstract topics studied in the book, no figures are included in it.

The first chapter is a short introduction giving some basic knowledge about Newtonian equations of motion, Hamiltonian formalism, and Liouville integrable systems. The main part of the book is contained in Chapter 2, which deals with one-dimensional motions of particles along a line or a circle. Liouville integrable problems are found by using the Lax pair technique. Several examples are obtained by solving three kinds of functional equations. From these integrable cases, other integrable systems are deduced by means of several transformations like change of variables, duplication and so on. Another method introduces a time dependant polynomial of degree N with respect to the variable X; the N zeros of this polynomial are interpreted as the positions of N particles (with the same masses) on a line. Assuming that the polynomial is solution of a linear partial derivative equation, several solvable many body problems are found. An extension of this approach is also presented, using the exact Lagrangian interpolation method. In Chapter 3, a generalized formulation of this last method is formulated for spaces of arbitrary dimensions. Chapter 4 deals with integrable systems in the plane. The method is based on the idea that integrable problems in two-dimensional space can be deduced from integrable problems in one-dimension assuming that the equations of motion hold also for complex values of the variables: it is the so called technique of complexification. A survey of integrable systems in the plane obtained by this method is given at the end of the chapter. The last chapter presents integrable many body systems in three-dimensional space. The idea is to identify some solvable evolution equations governing a time dependant matrix with the motion equations of a set of particles in three-dimensions. Again, several examples are presented.

As a conclusion, Classical Many-Body Problems Amenable to Exact Treatments…, is not intended for a very large audience, but it provides a rather exhaustive survey about integrable problems in some special cases of particle dynamics. It can be useful for researchers involved in celestial mechanics or can be used as a background for teaching an undergraduate course.