Preface ix
Chapter 1 Symplectic Manifolds 1
1.1 Introduction 1
1.2 Symplectic vector spaces 2
1.3 Symplectic manifolds 3
1.4 Vectors fields and flows 7
1.5 The Darboux theorem 22
1.6 Poisson brackets and Hamiltonian systems 25
1.7 Examples 31
1.8 Coadjoint orbits and their symplectic structures 35
1.9 Application to the group SO(n) 37
1.9.1 Application to the group SO(3) 39
1.9.2 Application to the group SO(4) 41
1.10 Exercises 47
Chapter 2 Hamiltonâ"Jacobi Theory 49
2.1 Eulerâ"Lagrange equation 49
2.2 Legendre transformation 52
2.3 Hamiltonâs canonical equations 53
2.4 Canonical transformations 55
2.5 Hamiltonâ"Jacobi equation 57
2.6 Applications 60
2.6.1 Harmonic oscillator 60
2.6.2 The Kepler problem 61
2.6.3 Simple pendulum 62
2.7 Exercises 64
Chapter 3 Integrable Systems 67
3.1 Hamiltonian systems and Arnoldâ"Liouville theorem 67
3.2 Rotation of a rigid body about a fixed point 75
3.2.1 The Euler problem of a rigid body 78
3.2.2 The Lagrange top 82
3.2.3 The Kowalewski spinning top 83
3.2.4 Special cases 90
3.3 Motion of a solid through ideal fluid 91
3.3.1 Clebschâs case 91
3.3.2 Lyapunovâ"Steklovâs case 93
3.4 Yangâ"Mills field with gauge group SU(2) 93
3.5 Appendix (geodesic flow and Eulerâ"Arnold equations) 95
3.6 Exercises 100
Chapter 4 Spectral Methods for Solving Integrable Systems 103
4.1 Lax equations and spectral curves 103
4.2 Integrable systems and Kacâ"Moody Lie algebras 104
4.3 Geodesic flow on SO(n) 107
4.4 The Euler problem of a rigid body 108
4.5 The Manakov geodesic flow on the group SO(4) 109
4.6 Jacobi geodesic flow on an ellipsoid and Neumann problem 114
4.7 The Lagrange top 115
4.8 Quartic potential, Garnier system 115
4.9 The coupled nonlinear Schr¶dinger equations 118
4.10 The Yangâ"Mills equations 119
4.11 The Kowalewski top 119
4.12 The Goryachevâ"Chaplygin top 121
4.13 Periodic infinite band matrix 122
4.14 Exercises 122
Chapter 5 The Spectrum of Jacobi Matrices and Algebraic Curves 129
5.1 Jacobi matrices and algebraic curves 129
5.2 Difference operators 135
5.3 Continued fraction, orthogonal polynomials and Abelian integrals 137
5.4 Exercises 140
Chapter 6 Griffiths Linearization Flows on Jacobians 143
6.1 Spectral curves 143
6.2 Cohomological deformation theory 144
6.3 Mittagâ"Leffler problem 148
6.4 Linearizing flows 149
6.5 The Toda lattice 150
6.6 The Lagrange top 153
6.7 Nahmâs equations 154
6.8 The n-dimensional rigid body 155
6.9 Exercises 156
Chapter 7 Algebraically Integrable Systems 159
7.1 Meromorphic solutions 159
7.2 Algebraic complete integrability 164
7.3 The Liouvilleâ"Arnoldâ"Adlerâ"van Moerbeke theorem 171
7.4 The Euler problem of a rigid body 173
7.5 The Kowalewski top 175
7.6 The H©nonâ"Heiles system 191
7.7 The Manakov geodesic flow on the group SO(4) 200
7.8 Geodesic flow on SO(4) with a quartic invariant 206
7.9 The geodesic flow on SO(n) for a left invariant metric 210
7.10 The periodic five-particle Kacâ"van Moerbeke lattice 212
7.11 Generalized periodic Toda systems 213
7.12 The Grossâ"Neveu system 214
7.13 The Kolossof potential 215
7.14 Exercises 215
Chapter 8 Generalized Algebraic Completely Integrable Systems 221
8.1 Generalities 221
8.2 The RDG potential and a five-dimensional system 225
8.3 The H©nonâ"Heiles problem and a five-dimensional system 229
8.4 The Goryachevâ"Chaplygin top and a seven-dimensional system 231
8.5 The Lagrange top 236
8.6 Exercises 237
Chapter 9 The Kortewegâ"de Vries Equation 241
9.1 Historical aspects and introduction 241
9.2 Stationary Schr¶dinger and integral Gelfandâ"Levitan equations 243
9.3 The inverse scattering method 255
9.4 Exercises 269
Chapter 10 KPâ"KdV Hierarchy and Pseudo-differential Operators 275
10.1 Pseudo-differential operators and symplectic structures 275
10.2 KdV equation, Heisenberg and Virasoro algebras 279
10.3 KP hierarchy and vertex operators 281
10.4 Exercises 290
References 293
Index 305
