经典可积系统导论

1 Introduction
2 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 model
3 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 solutions
4 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 top
5 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 systems
6 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 conditions
7 The Calogero-Moser model
　7.1 The spin Caloger0-Moser model
……
8 Isomonodromic deformations
9 Grassmannian and integrable hierarchies
10 The KP hierarchy
11 The KdV hierarchy
12 The Toda field theories
13 Classical inverse scattering method
14 Symplectic geometry
15 Riemann surfaces
16 Lie algebras
Index