Introduction to Hamiltonian dynamical systems and the N-body problem:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York u.a.
Springer
1992
|
| Ausgabe: | 1. print. |
| Schriftenreihe: | Applied mathematical sciences
90. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 292 S. graph. Darst. |
| ISBN: | 038797637X 354097637X |
Internformat
MARC
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| 100 | 1 | |a Meyer, Kenneth R. |d 1937- |e Verfasser |0 (DE-588)120262827 |4 aut | |
| 245 | 1 | 0 | |a Introduction to Hamiltonian dynamical systems and the N-body problem |c Kenneth R. Meyer ; Glen R. Hall |
| 250 | |a 1. print. | ||
| 264 | 1 | |a New York u.a. |b Springer |c 1992 | |
| 300 | |a XII, 292 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v 90. | |
| 650 | 0 | 7 | |a Vielkörperproblem |0 (DE-588)4078900-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hamiltonsches System |0 (DE-588)4139943-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hamiltonsches System |0 (DE-588)4139943-2 |D s |
| 689 | 0 | 1 | |a Vielkörperproblem |0 (DE-588)4078900-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Hall, Glen R. |e Verfasser |4 aut | |
| 830 | 0 | |a Applied mathematical sciences |v 90. |w (DE-604)BV000005274 |9 90 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-003845881 | ||
Datensatz im Suchindex
| _version_ | 1804120314327597056 |
|---|---|
| adam_text | Contents
Preface vii
Chapter I. Hamiltonian Differential Equations and the TV Body Problem 1
A. Background and Basic Definitions 1
Notation, Hamiltonian systems, Poisson bracket
B. Examples of Hamiltonian Systems 5
The harmonic oscillator, forced nonlinear oscillator, the elliptic
sine function, general Newtonian systems, a pair of harmonic
oscillators, linear flow on the torus, the Kirchhqff problem
C. The N Body Problem 17
The equations, the classical integrals, the Kepler problem, the
restricted 3 body problem
D. Simple Solutions 21
Central configurations, the Lagrangian equilateral triangle
solutions, the Euler Moulton collinear solutions, equilibria for
the restricted three body problem
E. Further Reading 28
Problems 29
Chapter II. Linear Hamiltonian Systems 33
A. Preliminaries 33
Hamiltonian symplectic matrices, reduction with a Lagrangian
set
B. Symplectic Linear Spaces 40
Symplectic basis, Lagrangian splitting
C. The Spectra of Hamiltonian and Symplectic Operators 44
Characteristic equations, J orthogonal, simple canonical forms
ix
x Contents
D. Nonelementary Divisors 50
General canonical forms
E. Periodic Systems and Floquet Lyapunov Theory 53
Multipliers, monodromy matrix, Liapunov transformation
F. Parametric Stability 56
Stability, parametric stability instability
G. The Critical Points in the Restricted Problem 59
Linear equations at Euler Lagrange points, Routh s critical
mass ratio, canonical forms
H. Further Reading 67
Appendix. Logarithm ofaSymplectic Matrix 67
Problems 69
Chapter III. Exterior Algebra and Differential Forms 72
A. Exterior Algebra 72
Multilinear, covector, k form, determinant
B. The Symplectic Form 76
Symplectic basis form, determinant of a symplectic matrix
C. Tangent Vectors and Cotangent Vectors 77
Contra co variant vectors, Einstein convention
D. Vector Fields and Differential Forms 79
Contra co variant vector fields, differential forms, exterior
derivative, Poincare s lemma
E. Changing Coordinates and Darboux s Theorem 83
Symplectic structure, symplectic coordinates, Darboux s
theorem
F. Integration and Stokes Theorem 85
Problems 85
Chapter IV. Symplectic Transformations and Coordinates 87
A. Symplectic Transformations 87
Definition, remainder function, variational equations
B. Applications 91
Rotating coordinates, Jacobi coordinates, (special case N = 2)
C. Differential Forms and Generating Functions 95
Symplectic form, action angle variables, d Alembert character,
generating functions, point transformations, Kepler s problem in
polar coordinates, the 3 body problem in Jacobi and Jacobi Polar
coordinates
D. Symplectic Transformations with Multipliers and Scaling. 102
Gravitational constant, Equations at an equilibrium point, the
restricted three body problem
E. Delaunay and Poincare Elements 104
F. Further Reading 107
Problems 107
Contents xi
Chapter V. Introduction to the Geometric Theory of Hamiltonian
Dynamical Systems 109
A. Introduction to Dynamical Systems 109
Orbit, trajectory, equilibrium point, period orbit, Hamiltonian
dynamical system, reparameterization
B. Discrete Dynamical Systems 113
Diffeomorphisms and symplectomorphisms, Henon map, time
x map, period map, isotopy, billiards table, crystal model
C. The Flow Box Theorem and Local Integrals 120
Ordinary point, local integrals, Hamiltonian flow box, integrals
in involution, ignorable coordinates
D. Noether s Theorem and Reduction 125
Discrete and continuous symmetry, reversible system, Noether s
Theorem
E. Periodic Solutions, Fixed Points, and Cross Sections 129
Elementary equilibrium points periodic orbits, multipliers,
cross section, Poincare map, systems with integrals, the cylinder
theorem
F. The Stable Manifold Theorem 136
Hyperbolic point, local and global stable and unstable manifold,
transversal homoclinic point, the X lemma, the Poincare tangle
G. Hyperbolic Systems 141
H. Further Reading 148
Appendix. Proof of Shadowing Lemma 149
Problems 152
Chapter VI. Continuation of Periodic Solutions 154
A. Continuation of Equilibrium Points and Periodic Solutions 155
B. Lyapunov s Center Theorem 156
Applications to the Euler and Lagrange libration points
C. Poincare s Orbits 158
D. Hill s Orbits 159
E. Comets 161
F. Continuation from the Restricted to the Full Problem 162
G. Some Elliptic Orbits 164
H. Further Reading 166
Problems 166
Chapter VII. Perturbation Theory and Normal Forms 168
A. The Method of Lie Transforms 168
Near identity change of coordinates, the forward Lie transform
algorithm, the remainder function
B. The Perturbation Algorithm 175
Example: Duff ing s equation, the general algorithm, the general
perturbation theorem, Duffing s equation revisited, uniqueness of
normal forms
xii Contents
C. Normal Form at an Equilibrium 182
The classic case, the general equilibria, example of normal forms
in the non simple case
D. Normal Form at i?4 189
E. Normal Forms for Periodic Systems and Diffeomorphisms 190
The reduction, general periodic case, general hyperbolic and
elliptic points, higher resonance in the planar case, normal forms
when multipliers are +1
F. Further Reading 199
Problems 199
Chapter VIII. Bifurcations of Periodic Orbits 201
A. Bifurcations of Periodic Solutions and Points. 201
Elementary fixed points, extremal fixed points, period doubling,
k bifurcation points.
B. Duffing Revisited. 212
Duffing at 1 1 resonance, k bifurcation in Duffing s equation
C. Schmidt s Bridges 216
D. Bifurcation of J?4 218
E. Further Reading 224
Problems 224
Chapter IX. Stability and KAM Theory 227
A. Elementary Stability Results 227
B. The Invariant Curve Theorem 229
C. A Simple Example—Duffing s Equation Again 232
D. Applications to the Restricted Problem 233
E. Arnold s Theorem 235
F. Stability of i?4 238
G. Further Reading 239
Problems 239
Chapter X. Twist Maps and Invariant Curves 241
A. Introduction 241
B. Notation and Definitions 242
C. Existence of Periodic Orbits 253
D. Monotone Orbits 255
E. Invariant Circles 266
F. Applications 274
G. Further Reading 274
Problems 277
References 279
Index 289
|
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| discipline | Physik Bauingenieurwesen Mathematik Vermessungswesen |
| edition | 1. print. |
| format | Book |
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| id | DE-604.BV006090037 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:40:07Z |
| institution | BVB |
| isbn | 038797637X 354097637X |
| language | English |
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| physical | XII, 292 S. graph. Darst. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Springer |
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| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Meyer, Kenneth R. 1937- Verfasser (DE-588)120262827 aut Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth R. Meyer ; Glen R. Hall 1. print. New York u.a. Springer 1992 XII, 292 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 90. Vielkörperproblem (DE-588)4078900-7 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Vielkörperproblem (DE-588)4078900-7 s DE-604 Hall, Glen R. Verfasser aut Applied mathematical sciences 90. (DE-604)BV000005274 90 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003845881&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Meyer, Kenneth R. 1937- Hall, Glen R. Introduction to Hamiltonian dynamical systems and the N-body problem Applied mathematical sciences Vielkörperproblem (DE-588)4078900-7 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| subject_GND | (DE-588)4078900-7 (DE-588)4139943-2 |
| title | Introduction to Hamiltonian dynamical systems and the N-body problem |
| title_auth | Introduction to Hamiltonian dynamical systems and the N-body problem |
| title_exact_search | Introduction to Hamiltonian dynamical systems and the N-body problem |
| title_full | Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth R. Meyer ; Glen R. Hall |
| title_fullStr | Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth R. Meyer ; Glen R. Hall |
| title_full_unstemmed | Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth R. Meyer ; Glen R. Hall |
| title_short | Introduction to Hamiltonian dynamical systems and the N-body problem |
| title_sort | introduction to hamiltonian dynamical systems and the n body problem |
| topic | Vielkörperproblem (DE-588)4078900-7 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| topic_facet | Vielkörperproblem Hamiltonsches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003845881&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
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