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Restricted Three-Body Problem and Holomorphic Curves 2018 ed. [Kõva köide]

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  • Formaat: Hardback, 374 pages, kõrgus x laius: 235x155 mm, kaal: 746 g, XI, 374 p., 1 Hardback
  • Sari: Pathways in Mathematics
  • Ilmumisaeg: 06-Sep-2018
  • Kirjastus: Birkhauser Verlag AG
  • ISBN-10: 3319722778
  • ISBN-13: 9783319722771
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Restricted Three-Body Problem and Holomorphic Curves 2018 ed.Suurem pilt
  • Formaat: Hardback, 374 pages, kõrgus x laius: 235x155 mm, kaal: 746 g, XI, 374 p., 1 Hardback
  • Sari: Pathways in Mathematics
  • Ilmumisaeg: 06-Sep-2018
  • Kirjastus: Birkhauser Verlag AG
  • ISBN-10: 3319722778
  • ISBN-13: 9783319722771
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319722771.html
Märksõnad:
The book serves as an introduction to holomorphic curves in symplectic manifolds, focusing on the case of four-dimensional symplectizations and symplectic cobordisms, and their applications to celestial mechanics.





The authors study the restricted three-body problem using recent techniques coming from the theory of pseudo-holomorphic curves.  The book starts with an introduction to relevant topics in symplectic topology and Hamiltonian dynamics before introducing some well-known systems from celestial mechanics, such as the Kepler problem and the restricted three-body problem. After an overview of different regularizations of these systems, the book continues with a discussion of periodic orbits and global surfaces of section for these and more general systems. The second half of the book is primarily dedicated to developing the theory of holomorphic curves - specifically the theory of fast finite energy planes - to elucidate the proofs of the existence results for global surfaces of section stated earlier. The book closes with a chapter summarizing the results of some numerical experiments related to finding periodic orbits and global surfaces of sections in the restricted three-body problem.

This book is also part of the Virtual Series on Symplectic Geometry





http://www.springer.com/series/16019
1 Introduction
1.1 The Birkhoff conjecture
1(1)
1.2 The power of holomorphic curves
2(2)
1.3 Systolic inequalities and symplectic embeddings
4(2)
1.4 Beyond the Birkhoff conjecture
6(3)
2 Symplectic Geometry and Hamiltonian Mechanics
2.1 Symplectic manifolds
9(1)
2.2 Symplectomorphisms
10(3)
2.2.1 Physical transformations
10(1)
2.2.2 The switch map
11(1)
2.2.3 Hamiltonian transformations
12(1)
2.3 Examples of Hamiltonians
13(7)
2.3.1 The free particle and the geodesic flow
13(2)
2.3.2 Stereographic projection and the geodesic flow of the round metric
15(1)
2.3.3 Mechanical Hamiltonians
16(2)
2.3.4 Magnetic Hamiltonians
18(1)
2.3.5 Physical symmetries
18(2)
2.3.6 Normal forms
20(1)
2.4 Hamiltonian structures
20(1)
2.5 Contact forms
21(2)
2.6 Liouville domains and contact type hypersurfaces
23(3)
2.7 Real Liouville domains and real contact manifolds
26(3)
3 Symmetries
3.1 Poisson brackets and Noether's theorem
29(3)
3.2 Hamiltonian group actions and moment maps
32(2)
3.3 Angular momentum, the spatial Kepler problem, and the Runge-Lenz vector
34(5)
3.3.1 Central force: conservation of angular momentum
34(1)
3.3.2 The Kepler problem and its integrals
35(1)
3.3.3 The Runge-Lenz vector: another integral of the Kepler problem
36(3)
3.4 Completely integrable systems
39(4)
3.5 The planar Kepler problem
43(4)
4 Regularization of Two-Body Collisions
4.1 Moser regularization
47(3)
4.2 The Levi-Civita regularization
50(2)
4.3 Ligon-Schaaf regularization
52(6)
4.3.1 Proof of some of the properties of the Ligon-Schaaf map
54(4)
5 The Restricted Three-Body Problem
5.1 The restricted three-body problem in an inertial frame
58(1)
5.2 Time-dependent transformations
59(2)
5.3 The circular restricted three-body problem in a rotating frame
61(2)
5.4 The five Lagrange points
63(7)
5.5 Hill's regions
70(1)
5.6 The rotating Kepler problem
71(1)
5.7 Moser regularization of the restricted three-body problem
72(5)
5.8 Hill's lunar problem
77(4)
5.8.1 Derivation of Hill's lunar problem
77(1)
5.8.2 Hill's lunar Hamiltonian
78(3)
5.9 Euler's problem of two fixed centers
81(4)
6 Contact Geometry and the Restricted Three-Body Problem
6.1 A contact structure for Hill's lunar problem
85(3)
6.2 Contact connected sum
88(4)
6.2.1 Contact version
90(2)
6.3 A real contact structure for the restricted three-body problem
92(1)
7 Periodic Orbits in Hamiltonian Systems
7.1 A short history of the research on periodic orbits
93(3)
7.2 Variational approach
96(3)
7.3 The kernel of the Hessian
99(5)
7.4 Periodic orbits of the first and second kind
104(7)
7.5 Symmetric periodic orbits and brake orbits
111(9)
7.6 Blue sky catastrophes
120(3)
7.7 Elliptic and hyperbolic orbits
123(4)
8 Periodic Orbits in the Restricted Three-Body Problem
8.1 Some heroes in the search for periodic orbits
127(2)
8.2 Periodic orbits in the rotating Kepler problem
129(9)
8.2.1 The shape of the orbits if E < 0
129(2)
8.2.2 The circular orbits
131(2)
8.2.3 The averaging method
133(3)
8.2.4 Periodic orbits of the second kind
136(2)
8.3 The retrograde and direct periodic orbit
138(11)
8.3.1 Low energies
138(2)
8.3.2 Birkhoff's shooting method
140(7)
8.3.3 The Birkhoff set
147(2)
8.4 Periodic orbits of the second kind for small mass ratios
149(2)
8.5 Lyapunov orbits
151(6)
8.6 Sublevel sets of a Hamiltonian and 1-handles
157(6)
9 Global Surfaces of Section
9.1 Disk-like global surfaces of section
163(3)
9.2 Obstructions
166(4)
9.3 Perturbative methods
170(3)
9.3.1 Global surface of section
172(1)
9.4 Existence results from holomorphic curve theory
173(4)
9.4.1 A simple example
173(3)
9.4.2 General results
176(1)
9.5 Invariant global surfaces of section
177(2)
9.6 Fixed points and periodic points
179(1)
9.7 Reversible maps and symmetric fixed points
180(3)
10 The Maslov Index
10.1 The Maslov index for loops
183(2)
10.2 The Maslov cycle
185(10)
10.3 The Maslov index for paths
195(1)
10.4 The Conley-Zehnder index
196(1)
10.5 Invariants of the group SL(2, R)
197(6)
10.6 The rotation number
203(4)
11 Spectral Flow
11.1 A Fredholm operator and its spectrum
207(8)
11.2 The spectrum bundle
215(6)
11.3 Winding numbers of eigenvalues
221(4)
12 Convexity
12.1 Convex hypersurfaces
225(5)
12.2 Convexity implies dynamical convexity
230(8)
12.3 Hamiltonian flow near a critical point of index 1
238(5)
13 Finite Energy Planes
13.1 Holomorphic planes
243(2)
13.2 The Hofer energy of a holomorphic plane
245(3)
13.3 The Omega-limit set of a finite energy plane
248(3)
13.4 Non-degenerate finite energy planes
251(1)
13.5 The asymptotic formula
252(4)
13.6 The index inequality and fast finite energy planes
256(9)
14 Siefring's Intersection Theory for Fast Finite Energy Planes
14.1 Positivity of intersection for closed curves
265(2)
14.2 The algebraic intersection number for finite energy planes
267(4)
14.3 Siefring's intersection number
271(1)
14.4 Siefring's inequality
272(7)
14.5 Computations and applications
279(6)
15 The Moduli Space of Fast Finite Energy Planes
15.1 Fredholm operators
285(6)
15.2 The first Chern number
291(4)
15.3 The normal Conley--Zehnder index
295(3)
15.4 An implicit function theorem
298(2)
15.5 Exponential weights
300(6)
15.6 Automatic transversality
306(2)
15.7 The R-quotient
308(3)
16 Compactness
16.1 Negatively punctured finite energy planes
311(2)
16.2 Weak SFT-compactness
313(1)
16.3 The systole
314(3)
16.4 Dynamical convexity
317(6)
17 Construction of Global Surfaces of Section
17.1 Open book decompositions
323(16)
17.1.1 More examples of finite energy planes
327(4)
17.1.2 Invariant surfaces of section and linking
331(2)
17.1.3 Global surface of section to open book
333(4)
17.1.4 Topological restrictions on open books
337(2)
18 Numerics and Dynamics via Global Surfaces of Section
18.1 Symmetric orbits
339(2)
18.2 Finding orbits via shooting
341(4)
18.2.1 Following an orbit by varying
342(1)
18.2.2 Conley--Zehnder index
342(1)
18.2.3 How to make the numerics rigorous
343(1)
18.2.4 Integration with error bounds
344(1)
18.2.5 Finding periodic orbits
345(1)
18.3 Numerical construction of a foliation by global surfaces of section
345(5)
18.3.1 Numerical holomorphic curves
345(2)
18.3.2 Some implementation details
347(1)
18.3.3 An ad hoc approach
348(2)
18.4 Finding a discretized return map and seeing the dynamics
350(7)
18.4.1 Some results and observations
351(4)
18.4.2 Another return map
355(2)
Bibliography 357(14)
Index 371