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BL
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| Record Number
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864344
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| Main Entry
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Frauenfelder, Urs
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| Title & Author
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The restricted three-body problem and holomorphic curves /\ Urs Frauenfelder, Otto van Koert.
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| Publication Statement
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Cham, Switzerland :: Birkhauser,, [2018]
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| Series Statement
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Pathways in Mathematics
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| Page. NO
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1 online resource (381 pages)
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| ISBN
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3319722786
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: 9783319722788
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3319722778
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9783319722771
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| Notes
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8.4 Periodic orbits of the second kind for small mass ratios
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Contents; Chapter 1 Introduction; 1.1 The Birkhoff conjecture; 1.2 The power of holomorphic curves; 1.3 Systolic inequalities and symplectic embeddings; 1.4 Beyond the Birkhoff conjecture; Chapter 2 Symplectic Geometry and Hamiltonian Mechanics; 2.1 Symplectic manifolds; 2.2 Symplectomorphisms; 2.2.1 Physical transformations; 2.2.2 The switch map; 2.2.3 Hamiltonian transformations; 2.3 Examples of Hamiltonians; 2.3.1 The free particle and the geodesic flow; 2.3.2 Stereographic projection and the geodesic flow of the round metric; 2.3.3 Mechanical Hamiltonians
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2.3.4 Magnetic Hamiltonians2.3.5 Physical symmetries; 2.3.6 Normal forms; 2.4 Hamiltonian structures; 2.5 Contact forms; 2.6 Liouville domains and contact type hypersurfaces; 2.7 Real Liouville domains and real contact manifolds; Chapter 3 Symmetries; 3.1 Poisson brackets and Noether's theorem; 3.2 Hamiltonian group actions and moment maps; 3.3 Angular momentum, the spatial Kepler problem, and the Runge-Lenz vector; 3.3.1 Central force: conservation of angular momentum; 3.3.2 The Kepler problem and its integrals; 3.3.3 The Runge-Lenz vector: another integral of the Kepler problem
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3.4 Completely integrable systems3.5 The planar Kepler problem; Chapter 4 Regularization of Two-Body Collisions; 4.1 Moser regularization; 4.2 The Levi-Civita regularization; 4.3 Ligon-Schaaf regularization; 4.3.1 Proof of some of the properties of the Ligon-Schaaf map; Chapter 5 The Restricted Three-Body Problem; 5.1 The restricted three-body problem in an inertial frame; 5.2 Time-dependent transformations; 5.3 The circular restricted three-body problem in a rotating frame; 5.4 The five Lagrange points; 5.5 Hill's regions; 5.6 The rotating Kepler problem
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5.7 Moser regularization of the restricted three-body problem5.8 Hill's lunar problem; 5.8.1 Derivation of Hill's lunar problem; 5.8.2 Hill's lunar Hamiltonian; 5.9 Euler's problem of two fixed centers; Chapter 6 Contact Geometry and the Restricted Three-Body Problem; 6.1 A contact structure for Hill's lunar problem; 6.2 Contact connected sum; 6.2.1 Contact version; 6.3 A real contact structure for the restricted three-body problem; Chapter 7 Periodic Orbits in Hamiltonian Systems; 7.1 A short history of the research on periodic orbits; 7.2 Variational approach; 7.3 The kernel of the Hessian
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7.4 Periodic orbits of the first and second kind7.5 Symmetric periodic orbits and brake orbits; 7.6 Blue sky catastrophes; 7.7 Elliptic and hyperbolic orbits; Chapter 8 Periodic Orbits in the Restricted Three-Body Problem; 8.1 Some heroes in the search for periodic orbits; 8.2 Periodic orbits in the rotating Kepler problem; 8.2.1 The shape of the orbits if; 8.2.2 The circular orbits; 8.2.3 The averaging method; 8.2.4 Periodic orbits of the second kind; 8.3 The retrograde and direct periodic orbit; 8.3.1 Low energies; 8.3.2 Birkhoff's shooting method; 8.3.3 The Birkhoff set
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| Abstract
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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.
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| Subject
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Hamiltonian operator.
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Holomorphic functions.
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Three-body problem.
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Hamiltonian operator.
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Holomorphic functions.
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MATHEMATICS-- Calculus.
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MATHEMATICS-- Mathematical Analysis.
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Three-body problem.
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| Dewey Classification
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515/.98
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| LC Classification
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QA331.F73 2018
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| Added Entry
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Koert, Otto van
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