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E-grāmata: Central Configurations, Periodic Orbits, and Hamiltonian Systems

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Central Configurations, Periodic Orbits, and Hamiltonian SystemsPalielināt
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The notes of this book originate from three series of lectures given at the Centre de Recerca Matemątica (CRM) in Barcelona. The first one is dedicated to the study of periodic solutions of autonomous differential systems in R n via the Averaging Theory and was delivered by Jaume Llibre. The second one, given by Richard Moeckel, focusses on methods for studying Central Configurations. The last one, by Carles Simó, describes the main mechanisms leading to a fairly global description of the dynamics in conservative systems.The book is directed towards graduate students and researchers interested in dynamical systems, in particular in the conservative case, and aims at facilitating the understanding of dynamics of specific models. The results presented and the tools introduced in this book include a large range of applications.

1 The Averaging Theory for Computing Periodic Orbits.- Introduction: the classical theory.- Averaging theory for arbitrary order and dimension.- Three applications of Theorem.- 2 Lectures on Central Configurations.- The n-body problem.- Symmetries and integrals.- Central configurations and self-similar solutions.- Matrix equations of motion.- Homographic motions of central configurations in Rd.- Albouy-Chenciner reduction and relative equilibria in Rd.- Homographic motions in Rd.- Central configurations as critical points.- Collinear central configurations.- Morse indices of non-collinear central configurations.- Morse theory for CC"s and SBC"s.- Dziobek configurations.- Convex Dziobek central configurations.- Generic finiteness for Dziobek central configurations.- Some open problems.- 3 Dynamical Properties of Hamiltonian Systems.- Introduction.- Low dimension.- Some theoretical results, their implementation and practical tools.- Applications to Celestial Mechanics.

Jaume Llibre is full professor at the Autonomous University of Barcelona (Spain) and a member of the Royal Academy of Sciences and Arts of Barcelona. He was a long-term visitor at different important universities and research institutes and is the author of many papers and some books. His main results deal with the periodic orbits either of different kinds of self-maps, or of vector fields, Hamiltonian systems and celestial mechanics.
1 The Averaging Theory for Computing Periodic Orbits
1(104)
Jaume Llibre
1.1 Preface
1(1)
1.2 Introduction: the classical theory
2(44)
1.2.1 A first order averaging method for periodic orbits
2(2)
1.2.2 Four applications
4(13)
1.2.3 Other first order averaging methods for periodic orbits
17(1)
1.2.4 Three applications
18(14)
1.2.5 Another first order averaging method for periodic orbits
32(3)
1.2.6 Proof of Theorem 1.2.1
35(9)
1.2.7 Proof of Theorem 1.2.9
44(1)
1.2.8 Proof of Theorem 1.2.18
45(1)
1.3 Averaging theory for arbitrary order and dimension
46(18)
1.3.1 Statement of the main results
47(4)
1.3.2 Proofs of Theorems 1.3.5 and 1.3.6
51(7)
1.3.3 Computing formulae
58(1)
1.3.4 Fifth order averaging of Theorem 1.3.5
59(1)
1.3.5 Fourth order averaging of Theorem 1.3.6
60(3)
1.3.6 Appendix: basic results on the Brouwer degree
63(1)
1.4 Three applications of Theorem 1.3.5
64(41)
1.4.1 The averaging theory of first, second and third order
64(2)
1.4.2 The Henon--Heiles Hamiltonian
66(6)
1.4.3 Limit cycles of polynomial differential systems
72(7)
1.4.4 The generalized polynomial differential Lienar equation
79(20)
Bibliography
99(6)
2 Central Configurations
105(64)
Richard Moeckel
2.1 The n-body problem
105(1)
2.2 Symmetries and integrals
106(2)
2.3 Central configurations and self-similar solutions
108(4)
2.4 Matrix equations of motion
112(4)
2.5 Homographic motions of central configurations in Rd
116(3)
2.6 Albouy--Chenciner reduction and relative equilibria in Rd
119(8)
2.7 Homographic motions in Rd
127(2)
2.8 Central configurations as critical points
129(9)
2.9 Collinear central configurations
138(6)
2.10 Morse indices of non-collinear central configurations
144(2)
2.11 Morse theory for CC's and SBC's
146(5)
2.12 Dziobek configurations
151(3)
2.13 Convex Dziobek central configurations
154(3)
2.14 Generic finiteness for Dziobek central configurations
157(5)
2.15 Some open problems
162(7)
Bibliography
165(4)
3 Dynamical Properties of Hamiltonian Systems with Applications to Celestial Dynamics
169(58)
Carles Simo
3.1 Introduction
169(4)
3.1.1 Continuous and discrete conservative systems
170(3)
3.1.2 Comments on the contents
173(1)
3.2 Low dimension
173(15)
3.2.1 The Henon map
174(8)
3.2.2 The standard map
182(2)
3.2.3 Return maps: the separatrix map
184(4)
3.3 Some theoretical results, their implementation and practical tools
188(13)
3.3.1 A preliminary tool: the integration of the ODE, Taylor method and jet transport
188(3)
3.3.2 Normal forms
191(2)
3.3.3 Stability results: KAM theory and related topics
193(3)
3.3.4 Invariant manifolds
196(2)
3.3.5 Instability, bounds and detection
198(3)
3.4 Applications to Celestial Mechanics
201(26)
3.4.1 An elementary mission around L1
202(3)
3.4.2 Escape and confinement in the Sitnikov problem
205(4)
3.4.3 Practical confinement around triangular points
209(6)
3.4.4 Infinitely many choreographies in the three-body problem
215(4)
3.4.5 Evidences of diffusion related to the centre manifold of L3
219(8)
Bibliography 227
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