Classical Many-Body Problems Amenable to Exact Treatments: (Solvable and/or Integrable and/or Linearizable…) in One-, Two- and Three-Dimensional Space

Description

cited on [EqWorld](http://eqworld.ipmnet.ru/)* * *This is a book by one of the pioneers of soliton theory, known also as the theory of integrable systems. In particular, one of the early monographs on the inverse scattering approach to KdV was written by the author and A. Degasperis [Spectral transform and solitons. Vol. I, Lecture Notes in Comput. Sci., 144, North-Holland, Amsterdam, 1982; [MR0680040](https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=680040&loc=fromrevtext)] and proved to be rather influential. One of the distinctive features of that book is shared also by the monograph under review: a great attention to all details of calculations, which, in principle, allows an undergraduate student or just a novice to follow them. Thus, the book combines the features of a scientific monograph and a textbook (or even the syllabus of a special university course). Calogero's name is firmly associated with the Calogero-Moser system, i.e. a one-dimensional many-particle system with pairwise interactions via a potential proportional to the inverse square of the relative displacement of the particles, or to the inverse square of the sin (or of the sinh) of this relative displacement. Since the discovery of these systems in the mid-1970s, a tremendous amount of literature has been devoted to them and to their generalizations. However, in the last few years the author's main attention has been devoted to finding and studying integrable cases of many-particle systems in higher dimensions, in particular, in the "physical'' three-dimensional space. The monograph at hand is devoted to a systematic presentation of all these findings. It consists of five chapters: Classical (nonquantal, nonrelativistic) many-body problems; One-dimensional systems. Motions on the line and on the circle; NN-body problems treatable via techniques of exact Lagrangian interpolation in spaces of one or more dimensions; Solvable and/or integrable many-body problems in the plane, obtained by complexification; Many-body systems in ordinary (three-dimensional) space: solvable, integrable, linearizable problems; as well as eight appendices. The short first chapters serve mainly to recall the basic notions of Hamiltonian mechanics and to fix various notions and notations. The longest chapter is the second one (about 300 pages). Despite its length, it could not and does not contain everything known today about systems of Calogero-Moser type. Rather, it contains everything discovered by Calogero himself (with co-workers), and a couple of results he considers congenial to his own work (basically, on the Ruijsenaars-Schneider systems). The topics include (to mention only some of them): various ansätze for Lax representations and the corresponding functional equations (this covers the CM systems, the RS systems, and the systems generalizing the one describing the peakon solutions of the Camassa-Holm equation); various techniques for finding explicit solutions (Olshanetskiĭ-Perelomov techniques, tricks related to the motion of zeroes of polynomials whose evolution is governed by linear equations, etc.) Mainly, rational and hyperbolic/trigonometric systems are considered; less attention is paid to elliptic systems. One should not look here for results on rr-matrices and Hamiltonian reduction, for generalizations related to general root systems or reflection groups, for the relation to gauge theories, or for other developments in which the author did not participate personally. Nevertheless, reading this chapter should enable anyone to enter this fascinating area by a careful working through of many other topics together with their discoverer. The topics of Chapters 3–5 are less popular (maybe because the findings presented there are much more recent), but the reader can be assured to also find there a large amount of fascinating mathematical models, tricks, observations, etc. Everything is written in great detail and very carefully; the text is almost free of misprints. One could doubt the physical applications of all this, but the mathematical value of these models is indisputable. All in all, the book describes part of the modern theory of integrable systems of classical mechanics as seen by one of its creators. It is highly accessible and will serve as a standard reference for a long time. Reviewed by [Yuri B. Suris](https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=241448)* * *This book focuses on exactly treatable classical (i.e. non-quantal non-relativistic) many-body problems, as described by Newton's equation of motion for mutually interacting point particles. Most of the material is based on the author's research and is published here for the first time in book form. One of the main novelties is the treatment of problems in two- and three-dimensional space. Many related techniques are presented, e.g. the theory of generalized Lagrangian-type interpolation in higher-dimensional spaces. This book is written for students as well as for researchers; it works out detailed examples before going on to treat more general cases. Many results are presented via exercises, with clear hints pointing to their solutions.