经典可积系统导论

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内容概要

作者：(法国) 贝博龙 (Babelon.O)

书籍目录

1 Introduction2 Integrable dynamical systems  2.1 Introduction  2.2 The Liouville theorem  2.3 Action-angle variables  2.4 Lax pairs  2.5 Existence of an r-matrix  2.6 Commuting flows  2.7 The Kepler problem  2.8 The Euler top  2.9 The Lagrange top  2.10 The Kowalevski top  2.11 The Neumann model  2.12 Geodesics on an ellipsoid  2.13 Separation of variables in the Neumann model3 Synopsis of integrable systems　3.1 Examples of Lax pairs with spectral parameter　3.2 The Zakharov-Shabat construction　3.3 Coadjoint orbits and Hamiltonian formalism　3.4 Elementary flows and wave function　3.5 Factorization problem　3.6 Tau.functions　3.7 Integrable field theories and monodromy matrix　3.8 Abelianization　3.9 Poisson brackets of the monodromy matrix　3.10 The group of dressing transformations  3.11 Soliton solutions4 Algebraic methods  4.1 The classical and modifiedⅥln9-Baxter equations  4.2 Algebraic meaning of the classical Yan9-Baxter equations　4.3 Adler-Kostant-Symes scheme  4.4 Construction of integrable systems  4.5 Solving by factorization  4.6 The open Toda chain　4.7 The r.matrix of the Toda models  4.8 Solution of the open Toda chain  4.9 Toda system and Hamiltonian reduction  4.10 The Lax pair of the Kowalevski top5 Analytical methods  5.1 The spectral curve  5.2 The eigenvector bundle  5.3 The adjoint linear system  5.4 Time evolution  5.5 Theta-functions formulae  5.6 Baker-Akhiezer functions  5.7 Linearization and the factorization problem  5.8 Tau-functions  5.9 Symplectic form  5.10 Separation of variables and the spectral curve  5.11 Action-angle variables  5.12 Riemann surfaces and integrability  5.13 The Kowalevski top  5.14 Infinite-dimensional systems6 The closed T0da chain  6.1 The model　6.2 The spectral curve　6.3 The eigenvectors　6.4 Reconstruction formula　6.5 Symplectic structure　6.6 The Sklyanin approach　6.7 The Poisson brackets　6.8 Reality conditions7 The Calogero-Moser model　7.1 The spin Caloger0-Moser model  ……8 Isomonodromic deformations9 Grassmannian and integrable hierarchies10 The KP hierarchy11 The KdV hierarchy12 The Toda field theories13 Classical inverse scattering method14 Symplectic geometry15 Riemann surfaces16 Lie algebrasIndex

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作者简介

《经典可积系统导论》是贝博龙编著的，provides a thorough introduction to the theory of classical integrable systems, discussing the various approaches to the subject and explaining their interrelations. The book begins by introducing the central ideas of the theory of integrable systems, based on Lax representations, loop groups and Riemann surfaces. These ideas are then illustrated with detailed studies of model systems. The connection between isomon- odromic deformation and integrability is discussed, and integrable field theories are covered in detail. The KP, KdV and 'lbda hierarchies are explained using the notion of Grassmannian, vertex operators and pseudo-differential operators. A chapter is devoted to the inverse scattering method and three complementary chapters cover the necessary mathematical tools from symplectic geometry, Riemann surfaces and Lie algebras.   The book contains many worked examples and is suitable for use as a textbook on graduate courses. It also provides a comprehensive reference for researchers already working in the field. O LIVIE R B A B E L O N has been a member of the Centre National de la Recherche Sci- entifique （CNRS） since 1978. He works at the Laboratoire de Physique Theorique et Hautes Energies （LPTRE） at the University of Paris VI-Paris VII. His main fields of interest are particle physics, gauge theories and integrables systems. M I C H E L T A L O N has been a member of the CNRS since 1977. He works at the LPTHE at the University of Paris VI-Paris VII. He is involved in the computation of radiative corrections and anomalies in gauge theories and integrable systems. DENIS BERNARD has been a member of the CNRS since 1988. He currently works at the Service de Physique Theorique de Saclay. His main fields of interest are conforma[ field theories and integrable systems, and other aspects of statistical field theories, including statistical turbulence.

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