Introduction to Hamiltonian dynamical systems and the N-body problem:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
New York [u.a.]
Springer
2009
|
| Edition: | 2. ed. |
| Series: | Applied mathematical sciences
90. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 399 S. Ill., graph. Darst. |
| ISBN: | 9780387097237 |
Staff View
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 ; Dan Offin |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2009 | |
| 300 | |a XIII, 399 S. |b Ill., 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 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Many-body problem | |
| 650 | 0 | 7 | |a Vielkörperproblem |0 (DE-588)4078900-7 |2 gnd |9 rswk-swf |
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| 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 | |
| 700 | 1 | |a Offin, Dan |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:BVB01-017168247 | ||
Record in the Search Index
| _version_ | 1804138689811447808 |
|---|---|
| adam_text | Contents
1. Hamiltonian Systems..................................... 1
1.1 Notation.............................................. 1
1.2
Hamilton s Equations...................................
2
1.3 The
Poisson
Bracket
....................................
З
1.4
The Harmonic Oscillator
................................ 5
1.5
The Forced Nonlinear Oscillator
.......................... 6
1.6
The Elliptic Sine Function
............................... 7
1.7
General Newtonian System
.............................. 9
1.8
A Pair of Harmonic Oscillators
........................... 10
1.9
Linear Flow on the Torus
................................ 14
1.10
Euler-Lagrange Equations
............................... 15
1.11
The Spherical Pendulum
................................ 21
1.12
The
Kirchhoff
Problem
.................................. 22
2.
Equations of Celestial Mechanics
......................... 27
2.1
The TV-Body Problem
................................... 27
2.1.1
The Classical Integrals
............................ 28
2.1.2
Equilibrium Solutions
............................. 29
2.1.3
Central Configurations
............................ 30
2.1.4
The Lagrangian Solutions
......................... 31
2.1.5
The Euier-Moulton Solutions
...................... 33
2.1.6
Total Collapse
................................... 34
2.2
The 2-Body Problem
.................................... 35
2.2.1
The Kepler Problem
.............................. 36
2.2.2
Solving the Kepler Problem
........................ 37
2.3
The Restricted 3-Body Problem
.......................... 38
2.3.1
Equilibria of the Restricted Problem
................ 41
2.3.2
Hill s Regions
.................................... 42
3.
Linear Hamiltonian Systems
.............................. 45
3.1
Preliminaries
........................................... 45
3.2
Symplectic Linear Spaces
................................ 52
3.3
The Spectra of Hamiltonian and Symplectic Operators
...... 56
3.4
Periodic Systems and Floquet-Lyapunov Theory
........... 63
χ
Contents
4.
Topics
in Linear
Theory
.................................. 69
4.1
Critical Points in the Restricted Problem
.................. 69
4.2
Parametric Stability
.................................... 78
4.3
Logarithm of a Symplectic Matrix
......................... 83
4.3.1
Functions of a Matrix
............................. 84
4.3.2
Logarithm of a Matrix
............................ 85
4.3.3
Symplectic Logarithm
............................. 87
4.4
Topology of Sp{2n,M)
...................................
88
4.5
Maslov Index and the Lagrangian Grassmannian
........... 91
4.6
Spectral Decomposition
................................. 99
4.7
Normal Forms for Hamiltonian Matrices
................... 103
4.7.1
Zero Eigenvalue
..................................103
4.7.2
Pure Imaginary Eigenvalues
.......................108
5.
Exterior Algebra and Differential Forms
..................117
5.1
Exterior Algebra
.......................................117
5.2
The Symplectic Form
...................................122
5.3
Tangent Vectors and Cotangent Vectors
...................122
5.4
Vector Fields and Differential Forms
......................125
5.5
Changing Coordinates and Darboux s Theorem
............129
5.6
Integration and Stokes Theorem
.........................131
6.
Symplectic Transformations
..............................133
6.1
General Definitions
.....................................133
6.1.1
Rotating Coordinates
.............................135
6.1.2
The Variational Equations
.........................136
6.1.3
Poisson
Brackets
.................................137
6.2
Forms and Functions
....................................138
6.2.1
The Symplectic Form
.............................138
6.2.2
Generating Functions
.............................138
6.2.3
Mathieu
Transformations
..........................140
6.3
Symplectic Scaling
......................................140
6.3.1
Equations Near an Equilibrium. Point
...............141
6.3.2
The Restricted 3-Body Problem
....................141
6.3.3
Hill s Lunar Problem
.............................143
7.
Special Coordinates
.......................................147
7.1
Jacobi Coordinates
.....................................147
7.1.1
The 2-Body Problem in Jacobi Coordinates
..........149
7.1.2
The 3-Body Problem in Jacobi Coordinates
..........150
7.2
Action-Angle Variables
.................................150
7.2.1
d Alembert Character
.............................151
7.3
General Action-Angle Coordinates
.......................152
7.4
Polar Coordinates
......................................154
7.4.1
Kepler s Problem in Polar Coordinates
..............155
Contents xi
7.4.2 The 3-Body Problem in Jacobi-Polar
Coordinates
___156
7.5
Spherical Coordinates
...................................157
7.6
Complex Coordinates
...................................160
7.6.1
Levi-Civita Regularization
........................161
7.7
Delaunay and
Poincaré
Elements
.........................163
7.7.1
Planar Delaunay Elements
.........................163
7.7.2
Planar
Poincaré
Elements
.........................165
7.7.3
Spatial Delaunay Elements
........................166
7.8
Pulsating Coordinates
...................................167
7.8.1
Elliptic Problem
..................................170
8.
Geometric Theory
........................................175
8.1
Introduction to Dynamical Systems
.......................175
8.2
Discrete Dynamical Systems
.............................179
8.2.1
Diffeomorphisms and Symplectomorphisms
.......... 179
8.2.2
The Henon Map
.................................. 181
8.2.3
The Time
τ
Map
................................. 182
8.2.4
The Period Map
................................. 182
8.2.5
The Convex Billiards Table
........................ 183
8.2.6
A Linear Crystal Model
........................... 184
8.3
The Flow Box Theorem
................................. 186
8.4
Noether s Theorem and Reduction
........................ 191
8.4.1
Symmetries Imply Integrals
........................191
8.4.2
Reduction
.......................................192
8.5
Periodic Solutions and Cross-Sections
.....................195
8.5.1
Equilibrium Points
...............................195
8.5.2
Periodic Solutions
................................196
8.5.3
A Simple Example
................................199
8.5.4
Systems with Integrals
............................200
8.6
The Stable Manifold Theorem
............................202
8.7
Hyperbolic Systems
.....................................208
8.7.1
Shift Automorphism and Subshifts of Finite Type
___208
8.7.2
Hyperbolic Structures
.............................210
8.7.3
Examples of Hyperbolic Sets
.......................211
8.7.4
The Shadowing Lemma
...........................213
8.7.5
The Conley-Smale Theorem
.......................213
9.
Continuation of Solutions
.................................217
9.1
Continuation Periodic Solutions
..........................217
9.2
Lyapunov Center Theorem
..............................219
9.2.1
Applications to the
Euler
and
Lagrange
points
.......220
9.3
Poincaré^s
Orbits
.......................................221
9.4
Hill s Orbits
...........................................222
9.5
Comets
...............................................224
9.6
From the Restricted to the Full Problem
..................225
xii Contents
9.7
Some Elliptic Orbits
....................................227
10.
Normal Forms
............................................231
10.1
Normal Form Theorems
.................................231
10.1.1
Normal Form at an Equilibrium Point
..............231
10.1.2
Normal Form at a Fixed Point
.....................234
10.2
Forward Transformations
................................237
10.2.1
Near-Identity Symplectic Change of Variables
........237
10.2.2
The Forward Algorithm
...........................238
10.2.3
The Remainder Function
..........................240
10.3
The Lie Transform Perturbation Algorithm
................243
10.3.1
Example: Dufimg s Equation
.......................243
10.3.2
The General Algorithm
...........................245
10.3.3
The General Perturbation Theorem
.................245
10.4
Normal Form at an Equilibrium
..........................250
10.5
Normal Form at £4
.....................................257
10.6
Normal Forms for Periodic Systems
.......................259
11.
Bifurcations of Periodic Orbits
...........................271
11.1
Bifurcations of Periodic Solutions
.........................271
11.1.1
Extremal Fixed Points
.............................273
11.1.2
Period Doubling
..................................274
11.1.3
¿-Bifurcation Points
..............................278
11.2
Duffing Revisited
.......................................282
11.2.1
^-Bifurcations in Duffing s Equation
................285
11.3
Schmidt s Bridges
......................................286
11.4
Bifurcations in the Restricted Problem
....................288
11.5
Bifurcation at £4
.......................................291
12.
Variational Techniques
....................................301
12.1
The JV-Body and the Kepler Problem Revisited
............302
12.2
Symmetry Reduction for Planar 3-Body Problem
...........305
12.3
Reduced Lagrangian Systems
............................308
12.4
Discrete Symmetry with Equal Masses
....................311
12.5
The Variational Principle
................................313
12.6
Isosceles 3-Body Problem
................................315
12.7
A Variational Problem for Symmetric Orbits
...............317
12.8
Instability of the Orbits and the Maslov Index
.............321
12.9
Remarks
..............................................327
13.
Stability and
KAM
Theory
...............................329
13.1
Lyapunov and Chetaev s Theorems
.......................331
13.2
Moser s Invariant Curve Theorem
........................335
13.3
Arnold s Stability Theorem
..............................338
13.4 1:2
Resonance
..........................................342
Contents xiii
13.5 1:3
Resonance
..........................................344
13.6 1:1
Resonance
..........................................346
13.7
Stability of Fixed Points
.................................349
13.8
Applications to the Restricted Problem
....................351
13.8.1
Invariant Curves for Small Mass
....................351
13.8.2
The Stability of Comet Orbits
.....................352
14.
Twist Maps and Invariant Circle
..........................355
14.1
Introduction
...........................................355
14.2
Notations and Definitions
................................356
14.3
Elementary Properties of Orbits
..........................360
14.4
Existence of Periodic Orbits
.............................366
14.5
The Aubry-Mather Theorem
............................370
14.5.1
A Fixed-Point Theorem
...........................370
14.5.2
Subsets of A
.....................................371
14.5.3
Nonmonotone
Orbits Imply Monotone Orbits
........374
14.6
Invariant Circles
........................................379
14.6.1
Properties of Invariant Circles
.....................379
14.6.2
Invariant Circles and Periodic Orbits
...............383
14.6.3
Relationship to the
KAM
Theorem
.................385
14.7
Applications
...........................................386
References
....................................................389
Index
.........................................................397
|
| any_adam_object | 1 |
| author | Meyer, Kenneth R. 1937- Hall, Glen R. Offin, Dan |
| author_GND | (DE-588)120262827 |
| author_facet | Meyer, Kenneth R. 1937- Hall, Glen R. Offin, Dan |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.39 |
| dewey-sort | 3515 239 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Bauingenieurwesen Mathematik Vermessungswesen |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV035364256 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:32:11Z |
| institution | BVB |
| isbn | 9780387097237 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017168247 |
| oclc_num | 298342882 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 DE-29T DE-91G DE-BY-TUM DE-20 |
| owner_facet | DE-355 DE-BY-UBR DE-11 DE-29T DE-91G DE-BY-TUM DE-20 |
| physical | XIII, 399 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| 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 ; Dan Offin 2. ed. New York [u.a.] Springer 2009 XIII, 399 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 90. Hamiltonian systems Many-body problem 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 Offin, Dan Verfasser aut Applied mathematical sciences 90. (DE-604)BV000005274 90 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017168247&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Meyer, Kenneth R. 1937- Hall, Glen R. Offin, Dan Introduction to Hamiltonian dynamical systems and the N-body problem Applied mathematical sciences Hamiltonian systems Many-body problem 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 ; Dan Offin |
| title_fullStr | Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth R. Meyer ; Glen R. Hall ; Dan Offin |
| title_full_unstemmed | Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth R. Meyer ; Glen R. Hall ; Dan Offin |
| 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 | Hamiltonian systems Many-body problem Vielkörperproblem (DE-588)4078900-7 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| topic_facet | Hamiltonian systems Many-body problem Vielkörperproblem Hamiltonsches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017168247&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT meyerkennethr introductiontohamiltoniandynamicalsystemsandthenbodyproblem AT hallglenr introductiontohamiltoniandynamicalsystemsandthenbodyproblem AT offindan introductiontohamiltoniandynamicalsystemsandthenbodyproblem |