The Restricted Three-Body Problem and Holomorphic Curves (eBook)

Contents 7
Chapter 1 Introduction 12
1.1 The Birkhoff conjecture 12
1.2 The power of holomorphic curves 13
1.3 Systolic inequalities and symplectic embeddings 15
1.4 Beyond the Birkhoff conjecture 17
Chapter 2 Symplectic Geometry and Hamiltonian Mechanics 20
2.1 Symplectic manifolds 20
2.2 Symplectomorphisms 21
2.2.1 Physical transformations 21
2.2.2 The switch map 22
2.2.3 Hamiltonian transformations 23
2.3 Examples of Hamiltonians 24
2.3.1 The free particle and the geodesic flow 24
2.3.2 Stereographic projection and the geodesic flow of the round metric 26
2.3.3 Mechanical Hamiltonians 27
2.3.4 Magnetic Hamiltonians 29
2.3.5 Physical symmetries 29
2.3.6 Normal forms 31
2.4 Hamiltonian structures 31
2.5 Contact forms 32
2.6 Liouville domains and contact type hypersurfaces 34
2.7 Real Liouville domains and real contact manifolds 37
Chapter 3 Symmetries 40
3.1 Poisson brackets and Noether’s theorem 40
3.2 Hamiltonian group actions and moment maps 43
3.3 Angular momentum, the spatial Kepler problem, and the Runge–Lenz vector 45
3.3.1 Central force: conservation of angular momentum 45
3.3.2 The Kepler problem and its integrals 46
3.3.3 The Runge–Lenz vector: another integral of the Kepler problem 47
3.4 Completely integrable systems 50
3.5 The planar Kepler problem 54
Chapter 4 Regularization of Two-Body Collisions 57
4.1 Moser regularization 57
4.2 The Levi-Civita regularization 60
4.3 Ligon–Schaaf regularization 62
4.3.1 Proof of some of the properties of the Ligon–Schaaf map 64
Chapter 5 The Restricted Three-Body Problem 67
5.1 The restricted three-body problem in an inertial frame 68
5.2 Time-dependent transformations 69
5.3 The circular restricted three-body problem in a rotating frame 71
5.4 The five Lagrange points 73
5.5 Hill’s regions 80
5.6 The rotating Kepler problem 81
5.7 Moser regularization of the restricted three-body problem 82
5.8 Hill’s lunar problem 87
5.8.1 Derivation of Hill’s lunar problem 87
5.8.2 Hill’s lunar Hamiltonian 88
5.9 Euler’s problem of two fixed centers 91
Chapter 6 Contact Geometry and the Restricted Three-Body Problem 95
6.1 A contact structure for Hill’s lunar problem 95
6.2 Contact connected sum 98
6.2.1 Contact version 100
6.3 A real contact structure for the restricted three-body problem 102
Chapter 7 Periodic Orbits in Hamiltonian Systems 103
7.1 A short history of the research on periodic orbits 103
7.2 Variational approach 106
7.3 The kernel of the Hessian 109
7.4 Periodic orbits of the first and second kind 114
7.5 Symmetric periodic orbits and brake orbits 121
7.6 Blue sky catastrophes 130
7.7 Elliptic and hyperbolic orbits 133
Chapter 8 Periodic Orbits in the Restricted Three-Body Problem 136
8.1 Some heroes in the search for periodic orbits 136
8.2 Periodic orbits in the rotating Kepler problem 138
8.2.1 The shape of the orbits if 138
8.2.2 The circular orbits 140
8.2.3 The averaging method 142
8.2.4 Periodic orbits of the second kind 145
8.3 The retrograde and direct periodic orbit 147
8.3.1 Low energies 147
8.3.2 Birkhoff’s shooting method 149
8.3.3 The Birkhoff set 156
8.4 Periodic orbits of the second kind for small mass ratios 158
8.5 Lyapunov orbits 160
8.6 Sublevel sets of a Hamiltonian and 1-handles 166
Chapter 9 Global Surfaces of Section 172
9.1 Disk-like global surfaces of section 172
9.2 Obstructions 175
9.3 Perturbative methods 179
9.3.1 Global surface of section 181
9.4 Existence results from holomorphic curve theory 182
9.4.1 A simple example 182
9.4.2 General results 185
9.5 Invariant global surfaces of section 186
9.6 Fixed points and periodic points 188
9.7 Reversible maps and symmetric fixed points 189
Chapter 10 The Maslov Index 192
10.1 The Maslov index for loops 192
10.2 The Maslov cycle 194
10.3 The Maslov index for paths 204
10.4 The Conley–Zehnder index 205
10.5 Invariants of the group 206
10.6 The rotation number 212
Chapter 11 Spectral Flow 216
11.1 A Fredholm operator and its spectrum 216
11.2 The spectrum bundle 224
11.3 Winding numbers of eigenvalues 230
Chapter 12 Convexity 234
12.1 Convex hypersurfaces 234
12.2 Convexity implies dynamical convexity 239
12.3 Hamiltonian flow near a critical point of index 1 247
Chapter 13 Finite Energy Planes 251
13.1 Holomorphic planes 251
13.2 The Hofer energy of a holomorphic plane 253
13.3 The Omega-limit set of a finite energy plane 256
13.4 Non-degenerate finite energy planes 259
13.5 The asymptotic formula 260
13.6 The index inequality and fast finite energy planes 264
Chapter 14 Siefring’s Intersection Theory for Fast Finite Energy Planes 273
14.1 Positivity of intersection for closed curves 273
14.2 The algebraic intersection number for finite energy planes 275
14.3 Siefring’s intersection number 279
14.4 Siefring’s inequality 280
14.5 Computations and applications 287
Chapter 15 The Moduli Space of Fast Finite Energy Planes 293
15.1 Fredholm operators 293
15.2 The first Chern number 299
15.3 The normal Conley–Zehnder index 303
15.4 An implicit function theorem 306
15.5 Exponential weights 308
15.6 Automatic transversality 314
15.7 The R-quotient 316
Chapter 16 Compactness 319
16.1 Negatively punctured finite energy planes 319
16.2 Weak SFT-compactness 321
16.3 The systole 322
16.4 Dynamical convexity 325
Chapter 17 Construction of Global Surfaces of Section 330
17.1 Open book decompositions 330
17.1.1 More examples of finite energy planes 334
17.1.2 Invariant surfaces of section and linking 338
17.1.3 Global surface of section to open book 340
17.1.4 Topological restrictions on open books 344
Chapter 18 Numerics and Dynamics via Global Surfaces of Section 346
18.1 Symmetric orbits 346
18.2 Finding orbits via shooting 348
18.2.1 Following an orbit by varying 349
18.2.2 Conley–Zehnder index 349
18.2.3 How to make the numerics rigorous 350
18.2.4 Integration with error bounds 351
18.2.5 Finding periodic orbits 352
18.3 Numerical construction of a foliation by global surfaces of section 352
18.3.1 Numerical holomorphic curves 352
18.3.2 Some implementation details 354
18.3.3 An ad hoc approach 355
Numerical construction of p?,c 356
Checking the conjectural formulas 356
18.4 Finding a discretized return map and seeing the dynamics 357
18.4.1 Some results and observations 358
Annulus maps 358
Graphs of disk maps 358
Collision orbits 360
18.4.2 Another return map 362
Bibliography 364
Index 378