Introduction to the Perturbation Theory of Hamiltonian Systems (eBook)

Dmitry Treschev, Oleg Zubelevich (Autoren)

eBook Download: PDF

2009 | 2010
X, 211 Seiten
Springer Berlin (Verlag)
978-3-642-03028-4 (ISBN)

Infos

This book is an extended version of lectures given by the ?rst author in 1995-1996 at the Department of Mechanics and Mathematics of Moscow State University. We believe that a major part of the book can be regarded as an additional material to the standard course of Hamiltonian mechanics. In comparison with the original Russian 1 version we have included new material, simpli?ed some proofs and corrected m- prints. Hamiltonian equations ?rst appeared in connection with problems of geometric optics and celestial mechanics. Later it became clear that these equations describe a large classof systemsin classical mechanics,physics,chemistry,and otherdomains. Hamiltonian systems and their discrete analogs play a basic role in such problems as rigid body dynamics, geodesics on Riemann surfaces, quasi-classic approximation in quantum mechanics, cosmological models, dynamics of particles in an accel- ator, billiards and other systems with elastic re?ections, many in?nite-dimensional models in mathematical physics, etc. In this book we study Hamiltonian systems assuming that they depend on some parameter (usually?), where for?= 0 the dynamics is in a sense simple (as a rule, integrable). Frequently such a parameter appears naturally. For example, in celestial mechanics it is accepted to take? equal to the ratio: the mass of Jupiter over the mass of the Sun. In other cases it is possible to introduce the small parameter ar- ?cially.

Inhalt

Preface 6
Contents 10
Hamiltonian Equations 12
Hamiltonian Systems: Geometric Point of View 12
Symplectic Maps 19
Inclusion of a Diffeomorphism into a Flow 24
The Classical Perturbation Theory 29
Introduction to the KAM Theory 33
The Kolmogorov Theorem 33
A Reduction of Theorems 2.2-2.4 to the Standard Version 36
Lower-Dimensional Tori 39
Hyperbolic Tori 45
Hyperbolic Tori: Weak Reducibility 50
Applications of the KAM Theory 54
Perturbations of a Quasi-Periodic Flow on a Torus 60
Proof of the Theorem on the Reduction of a Flow 63
Splitting of Asymptotic Manifolds 69
Normal Coordinates 69
The Poincaré-Melnikov Method: Traditional Aspect 71
The Poincaré-Melnikov Method: Multidimensional Aspect 78
The Separatrix Map 85
Definition and Formulas 85
Two-Dimensional Symplectic Map 85
Hamiltonian System with One and a Half Degrees of Freedom 88
Proof of Theorem 4.1 90
Poincaré-Melnikov Integral 94
Hamiltonian System 98
Separatrix Map for a Pendulum 100
Some Generalizations 101
Width of the Stochastic Layer 103
Definitions and Results 103
Theorems on the Stochastic Layer 106
Stochastic Layer for a Pendulum 110
KAM Theory and the Birkhoff Theorem 111
The Continuous Averaging Method 117
Description of the Method 117
Majorants 120
An Inclusion of a Map into a Flow 122
The Case of Arbitrary Algebra chi 126
Fast Phase Averaging 128
Analytic Properties of the Averaging Procedure 132
The Anti-Integrable Limit 141
Perturbation of the Standard Map 141
A General Construction 143
Further Examples 147
Anti-Integrable Limit in the Separatrix Map 150
Hill's Formula 153
General Remarks 153
Discrete Case 154
Continuous Case 163
Appendix 173
Diophantine Frequencies 173
Closures of Asymptotic Curves 178
Invariant Tori in a Neighborhood of a Resonance 187
Stability Islands in Separatrix Lobes 195
Elements of Analysis on Scales 202
Locally Convex Spaces 202
Scales of Banach Spaces 204
Other Examples of Scales 206
Majorant Functions 209
References 211
Index 219

Produkt-Details

• Erscheint lt. Verlag: 8.10.2009
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: X, 211 p. 20 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Dynamics • Hamiltonian Dynamics • Hamiltonian System • KAM Theory • Kolmogorov–Arnold–Moser theorem • manifold • Mechanics • perturbation theory
• ISBN-10: 3-642-03028-9 / 3642030289
• ISBN-13: 978-3-642-03028-4 / 9783642030284