Chapter Ⅰ The Beginning 1
1 Overview 1
Chapter Ⅱ Classical Background 19
2 Hamiltonian Formalism 19
3 Liouville Integrable Systems 27
4 Birkhoff Integrable Systems 34
5 KAM Theory 39
Chapter Ⅲ Birkhoff Coordinates 51
6 Background and Results 51
7 Actions 63
8 Angles 69
9 Cartesian Coordinates 74
10 Orthogonality Relations 85
11 The Diffeomorphism Property 91
12 The Symplectomorphism Property 102
Chapter Ⅳ Perturbed KdV Equations13 The Main Theorems 111
14 Birkhoff Normal Form 118
15 Global Coordinates and Frequencies 127
16 The KAM Theorem 133
17 Proof of the Main Theorems 139
Chapter Ⅴ The KAM Proof 145
18 Set Up and Summary of Main Results 145
19 The Linearized Equation 152
20 The KAM Step 160
21 Iteration and Convergence 165
22 The Excluded Set of Parameters 171
Chapter Ⅵ Kuksin's Lemma 177
23 Kuksin's Lemma 177
Chapter Ⅶ Background Material 187
A Analyticity 187
B Spectra 194
C KdV Hierarchy 207
Chapter Ⅷ Psi-Functions and FrequenciesD Construction of the Psi-Functions 211
E A Trace Formula 223
F Frequencies 227
Chapter Ⅸ Birkhoff Normal FormsG Two Results on Birkhoff Normal Forms 233
H Birkhoff Normal Form of Order 6 240
I Kramer's Lemma 248
J Nondegeneracy of the Second KdV Hamiltonian 252
Chapter Ⅹ Some Technicalities 257
K Symplectic Formalism 257
L Infinite Products 260
M Auxiliary Results 262
References 267
Index 275
Notations 57
1 a-cycles 57
2 Signs of ？ 62
3 Signs of ？ for real q 62
4 Labeling of periodic eigenvalues as q varies 64
5 Isolating neighbourhoods 65
6 A generic △-function 198
7 The set ？a,b？ 202
8 a-and b-cycles for N=2 224
9 a′-and b-cycles with basepoint λ0 for N=2 224
10 Signs of ？ for real q 282