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Symplectic techniques in physics. (English) Zbl 0576.58012
Cambridge etc.: Cambridge University Press. XI, 468 p. £32.50; $ 49.50 (1984).
In this book the authors show how symplectic geometry can be used for both the formulation of physical laws and the solution of arising problems. Regarding the topics from symplectic geometry needed for these purposes, this book is self-contained. The long introductory chapter, which has much in common with the first chapter of ”Geometric asymptotics” (1977; Zbl 0364.53011) by the same authors, is aimed at outlining the natural symplectic structures of optics, classical and quantum mechanics, classical field theory and their interconnections.
The central topic of the second chapter is the Hamiltonian actions of Lie groups on symplectic manifolds and the corresponding moment maps. The description of homogeneous symplectic manifolds, convexity properties of moment map, reductions of symplectic manifolds, stationary phase formula (due to Heckman-Duistermaat) are the main mathematical results of this chapter. Here the authors also consider the methods of choice of Hamiltonians for the collective motion, Euler equations on semidirect products, symplectic structure for Hartree-Fock approximations as the examples of physical applications.
The third chapter is devoted to the study of the formulation and solution of the equations of motion of particles in the presence of Yang-Mills field. The authors use their principle of general covariance as the conceptual foundation and the tool for justification of the equations arising. A local normal form for Hamiltonian actions of a compact Lie group and some relevant symplectic techniques (symplectic induction, isotropic and coisotropic embeddings) are also considered.
The fourth chapter deals with the group-theoretical technique for completely integrable systems (mainly finite dimensional). After the description of Arnold-type results on action-angle variables for completely integrable systems, the authors consider some ”industrial” methods for the construction of the families of integrals in involution: collective complete integrability, involution theorems of Mishchenko- Fomenko, Ratiu, Adler-Kostant. The concrete examples of completely integrable systems are: nonperiodic Toda lattice, Calogero type systems, rigid body motion. An introduction to Gel’fand-Dikij calculus and to the higher order calculus of variations with the application to the Korteweg- de Vries equation is also included. In the last chapter some results from Lie algebra theory with the applications to the deformations of homogeneous symplectic spaces are given.
This book is extremely well written and (unlike ”Geometric asymptotics”) is quite accessible to physicists, applied mathematicians and system theoreticians who have the usual background in differential manifolds- vector fields-differential forms. There are some omissions which seem to be inevitable due to the very fast progress in applications of symplectic geometry. The most important of them is the interaction of symplectic and algebraic geometry in studying completely integrable systems. Nevertheless from the reviewer’s point of view this book is one of the most important contributions, made by pure mathematicians to the applied areas within the past few years.
Reviewer: L.E.Fajbusovich

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
57R30 Foliations in differential topology; geometric theory
81T08 Constructive quantum field theory
53C80 Applications of global differential geometry to the sciences
53D50 Geometric quantization
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
57S25 Groups acting on specific manifolds
81V45 Atomic physics