A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.

This book is designed to expose from a general and universal standpoint a variety ofmethods and results concerning integrable systems ofclassical me- chanics. By such systems we mean Hamiltonian systems with a finite number of degrees of freedom possessing sufficiently many conserved quantities (in- tegrals ofmotion) so that in principle integration ofthe correspondingequa- tions of motion can be reduced to quadratures, i.e. to evaluating integrals of known functions. The investigation of these systems was an important line ofstudy in the last century which, among other things, stimulated the appearance of the theory ofLie groups. Early in our century, however, the work ofH. Poincare made it clear that global integrals of motion for Hamiltonian systems exist only in exceptional cases, and the interest in integrable systems declined. Until recently, only a small number ofsuch systems with two or more de- grees of freedom were known. In the last fifteen years, however, remarkable progress has been made in this direction due to the invention by Gardner, Greene, Kruskal, and Miura [GGKM 19671 ofa new approach to the integra- tion ofnonlinear evolution equations known as the inverse scattering method or the method of isospectral deformations. Applied to problems of mechanics this method revealed the complete in- tegrability of numerous classical systems. It should be pointed out that all systems of this kind discovered so far are related to Lie algebras, although often this relationship is not sosimpleas the oneexpressed by the well-known theorem of E. Noether.

This volume of the Encyclopaedia contains three contributions in the field of complex analysis. The topics treated are mean periodicity and convolutionequations, Yang-Mills fields and the Radon-Penrose transform, and stringtheory. The latter two have strong links with quantum field theory and the theory of general relativity. In fact, the mathematical results described inthe book arose from the need of physicists to find a sound mathematical basis for their theories. The authors present their material in the formof surveys which provide up-to-date accounts of current research. The book will be immensely useful to graduate students and researchers in complex analysis, differential geometry, quantum field theory, string theoryand general relativity.

This volume of the Encyclopaedia contains three contributions in the field of complex analysis; on mean periodicity and convolutionequations, Yang-Mills fields and the Radon-Penrose transform, and stringtheory. It is immensely useful to graduate students and researchers in complex analysis, differential geometry, quantum field theory, string theory and general relativity.

This monograph treats an extensively developed field in modern mathematical physics - the theory of generalized coherent states and their applications to various physical problems. Coherent states, introduced originally by Schrodinger and von Neumann, were later employed by Glauber for a quantal description of laser light beams. The concept was generalized by the author for an arbitrary Lie group. In the last decade the formalism has been widely applied to various domains of theoretical physics and mathematics. The area of applications of generalized coherent states is very wide, and a comprehensive exposition of the results in the field would be helpful. This monograph is the first attempt toward this aim. My purpose was to compile and expound systematically the vast amount of material dealing with the coherent states and available through numerous journal articles. The book is based on a number of undergraduate and postgraduate courses I delivered at the Moscow Physico-Technical Institute. In its present form it is intended for professional mathematicians and theoretical physicists; it may also be useful for university students of mathematics and physics. In Part I the formalism is elaborated and explained for some of the simplest typical groups. Part II contains more sophisticated material; arbitrary Lie groups and symmetrical spaces are considered. A number of examples from various areas of theoretical and mathematical physics illustrate advantages of this approach, in Part III. It is a pleasure for me to thank Dr. Yu. Danilov for many useful remarks.

A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.