Introduction to Hamiltonian dynamical systems and the N-body problem:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2017]
|
| Edition: | Third edition |
| Series: | Applied mathematical sciences
volume 90 |
| Subjects: | |
| Online Access: | http://www.springer.com/ Inhaltstext Inhaltsverzeichnis |
| Physical Description: | xiii, 384 Seiten Illustrationen |
| ISBN: | 9783319536903 3319536907 |
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| 245 | 1 | 0 | |a Introduction to Hamiltonian dynamical systems and the N-body problem |c Kenneth Meyer, Dan Offin |
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Record in the Search Index
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| adam_text |
Titel: Introduction to Hamiltonian dynamical systems and the N-Body problem
Autor: Meyer, Kenneth R
Jahr: 2017
Contents
1. Beginnings 1
1.1 Hamiltonian Equations.1
1.2 The Poisson Bracket.3
1.3 Harmonic Oscillator.4
1.4 Flows on a Sphere.6
1.5 Dense Flow on the Torus.14
1.6 Lemniscate Functions.15
1.7 Forced Nonlinear Oscillator.17
1.8 Newtonian System.18
1.9 Euler-Lagrange Equations.18
1.10 Spherical Pendulum .24
1.11 Problems.26
2. Hamiltonian Systems 29
2.1 Linear Equations.29
2.2 Symplectic Linear Spaces.36
2.3 Canonical Forms.41
2.4 Sp( 2,R).47
2.5 Floquet-Lyapunov Theory.50
2.6 Symplectic Transformations.52
2.6.1 The Variational Equations.55
2.6.2 Poisson Brackets.56
2.7 Symplectic Scaling.57
2.7.1 Equations Near an Equilibrium Point.58
2.8 Problems.58
3. Celestial Mechanics 61
3.1 The V-Body Problem.61
3.2 The 2-Body Problem .62
3.3 The Kepler Problem.64
3.4 Symmetries and Integrals.66
3.4.1 Symmetries.67
ix
X Contents
3.4.2 The Classical Integrals.67
3.4.3 Noether's Theorem.69
3.4.4 Reduction.70
3.4.5 Foundations.73
3.5 Equilibrium and Central Configurations.73
3.5.1 Equilibrium Solutions.73
3.5.2 Central Configurations.74
3.5.3 Rotating Coordinates.78
3.6 Total Collapse.79
3.7 Problems.80
4. The Restricted Problem 83
4.1 Defining. 83
4.2 Discrete Symmetry. 86
4.3 Equilibria of the Restricted Problem. 87
4.4 Hill's Regions. 89
4.5 Spectrum at the Equilibrium Points . 90
4.6 Mass Ratios. 93
4.7 Canonical Forms for the Matrix at £4 . 94
4.8 Other Restricted Problems . 99
4.8.1 Hill's Lunar Equations. 99
4.8.2 Elliptic Restricted Problem.101
4.9 Problems.102
5. Topics in Linear Theory 103
5.1 Parametric Stability.103
5.2 Logarithm of a Symplectic Matrix.108
5.2.1 Functions of a Matrix.109
5.2.2 Logarithm of a Matrix.110
5.2.3 Symplectic Logarithm.112
5.3 Spectral Decomposition.113
5.4 Normal Forms for Hamiltonian Matrices.117
5.4.1 Zero Eigenvalue.117
5.4.2 Pure Imaginary Eigenvalues.122
5.5 Topology of Sp(2n, R).129
5.5.1 Angle Function .129
5.5.2 Fundamental Group.131
5.6 Maslov Index.133
5.7 Problems.141
6. Local Geometric Theory 143
6.1 The Dynamical System Point of View.143
6.2 Discrete Dynamical Systems.147
6.2.1 Diffeomorphisms and Symplectomorphisms.147
6.2.2 The Time r-Map .149
6.2.3 The Period Map.149
Contents xi
6.3 Flow Box Theorems .151
6.4 Periodic Solutions and Cross Sections.155
6.4.1 Equilibrium Points.155
6.4.2 Periodic Solutions.156
6.4.3 A Simple Example .158
6.4.4 Systems with Integrals.159
6.5 The Stable Manifold Theorem .161
6.6 Problems.167
7. Symplectic Geometry 169
7.1 Exterior Algebra.169
7.1.1 Linear Symplectic Form.174
7.2 Tangent and Cotangent Spaces.174
7.3 Vector Fields and Differential Forms.177
7.3.1 Poincaré's Lemma.181
7.3.2 Changing Variables.182
7.4 Symplectic Manifold.183
7.4.1 Darboux's Theorem .185
7.5 Lie Groups.185
7.6 Group Actions.187
7.7 Moment Maps and Reduction.188
7.8 Integrable Systems .190
7.9 Problems.192
8. Special Coordinates 195
8.1 Differential Forms and Generating Functions.195
8.1.1 The Symplectic Form.195
8.1.2 Generating Functions.196
8.1.3 Mathieu Transformations.197
8.2 Jacobi Coordinates.197
8.3 Action-Angle Variables .200
8.3.1 d'Alembert Character.201
8.4 General Action-Angle Coordinates .202
8.5 Lemniscate Coordinates.205
8.6 Polar Coordinates.206
8.6.1 Kepler's Problem in Polar Coordinates.206
8.6.2 The 3-Body Problem in Jacobi-Polar
Coordinates .207
8.7 Spherical Coordinates.208
8.8 Complex Coordinates.211
8.8.1 Levi-Civita Regularization.212
8.9 Delaunay and Poincaré Elements.214
8.9.1 Planar Delaunay Elements.214
8.9.2 Planar Poincaré Elements.216
8.9.3 Spatial Delaunay Elements.217
xii Contents
8.10 Pulsating Coordinates .219
8.10.1 The Elliptic Restricted 3-Body Problem.221
8.11 Problems.223
9. Poincaré's Continuation Method 225
9.1 Continuation of Solutions.225
9.2 Lyapunov Center Theorem.227
9.2.1 Lyapunov Families at the Euler and Lagrange
points .228
9.3 Poincaré's Orbits.229
9.4 Hill's Orbits.230
9.5 Comets .232
9.6 From the Restricted to the Full Problem.233
9.7 Some Elliptic Orbits.236
9.8 Problems.237
10. Normal Forms 239
10.1 Normal Form Theorems.239
10.1.1 Normal Form at an Equilibrium Point .239
10.1.2 Normal Form at a Fixed Point.242
10.2 Forward Transformations.245
10.2.1 Near-Identity Symplectic Change of Variables.245
10.2.2 The Forward Algorithm.246
10.2.3 The Remainder Function.248
10.3 The Lie Transform Perturbation Algorithm.251
10.3.1 Example: Duffing's Equation.251
10.3.2 The General Algorithm.253
10.3.3 The General Perturbation Theorem.253
10.4 Normal Form at an Equilibrium.258
10.5 Normal Form at £4.266
10.6 Normal Forms for Periodic Systems.267
10.7 Problems.276
11. Bifurcations of Periodic Orbits 279
11.1 Bifurcations of Periodic Solutions.279
11.1.1 Elementary Fixed Points.280
11.1.2 Extremal Fixed Points.281
11.1.3 Period Doubling .283
11.1.4 fc-Bifurcation Points.287
11.2 Schmidt's Bridges.290
11.3 Bifurcations in the Restricted Problem.292
11.4 Hamiltoriian- Hopf Bifurcation .295
11.5 Problems.301
Contents xiii
12. Stability and KAM Theory 305
12.1 Lyapunov and Chetaev's Theorems.307
12.2 Local Geometry.311
12.3 Moser's Invariant Curve Theorem .312
12.4 Morris' Boundedness Theorem.315
12.5 Arnold's Stability Theorem.316
12.6 Singular Reduction.320
12.7 2:-l Resonance.330
12.8 3:-l Resonance.332
12.9 1:-1 Resonance.334
12.10 Stability of Fixed Points .339
12.11 Applications to the Restricted Problem .341
12.11.1 Invariant Curves for Small Mass.341
12.11.2 The Stability of Comet Orbits .341
12.12 Problems.343
13. Variational Techniques 345
13.1 The IV-Body and the Kepler Problem Revisited.347
13.2 Symmetry Reduction for Planar 3-Body Problem.349
13.3 Reduced Lagrangian Systems.352
13.4 Discrete Symmetry with Equal Masses.356
13.5 The Variational Principle.357
13.6 Isosceles 3-Body Problem.360
13.7 A Variational Problem for Symmetric Orbits.361
13.8 Instability of the Orbits and the Maslov Index.365
13.9 Remarks.371
References 373
Index 381 |
| any_adam_object | 1 |
| author | Meyer, Kenneth R. 1937- Offin, Dan |
| author_GND | (DE-588)120262827 |
| author_facet | Meyer, Kenneth R. 1937- Offin, Dan |
| author_role | aut aut |
| author_sort | Meyer, Kenneth R. 1937- |
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| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Bauingenieurwesen Mathematik Vermessungswesen |
| edition | Third edition |
| format | Book |
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| spelling | Meyer, Kenneth R. 1937- (DE-588)120262827 aut Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth Meyer, Dan Offin Third edition Cham Springer [2017] © 2017 xiii, 384 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences volume 90 Hamiltonian systems Many-body problem Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Vielkörperproblem (DE-588)4078900-7 gnd rswk-swf Dynamical Systems Hamiltonian Matrices Hamiltonian Systems KAM Theory Periodic Solutions Restricted 3-body Problem Symplectic Variational Methods Hamiltonsches System (DE-588)4139943-2 s Vielkörperproblem (DE-588)4078900-7 s DE-604 Offin, Dan aut Springer International Publishing (DE-588)1064344704 pbl Elektronische Reproduktion 9783319536910 Erscheint auch als Online-Ausgabe Meyer, Kenneth Introduction to Hamiltonian Dynamical Systems and the N-Body Problem Cham : Springer International Publishing, 2017 978-3-319-53691-0 Applied mathematical sciences volume 90 (DE-604)BV000005274 90 http://www.springer.com/ Verlag X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=55f13e78ab2741279d9a09a45e27b7ca&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029756709&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Meyer, Kenneth R. 1937- Offin, Dan Introduction to Hamiltonian dynamical systems and the N-body problem Applied mathematical sciences Hamiltonian systems Many-body problem Hamiltonsches System (DE-588)4139943-2 gnd Vielkörperproblem (DE-588)4078900-7 gnd |
| subject_GND | (DE-588)4139943-2 (DE-588)4078900-7 |
| 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 Meyer, Dan Offin |
| title_fullStr | Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth Meyer, Dan Offin |
| title_full_unstemmed | Introduction to Hamiltonian dynamical systems and the N-body problem Kenneth Meyer, 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 Hamiltonsches System (DE-588)4139943-2 gnd Vielkörperproblem (DE-588)4078900-7 gnd |
| topic_facet | Hamiltonian systems Many-body problem Hamiltonsches System Vielkörperproblem |
| url | http://www.springer.com/ http://deposit.dnb.de/cgi-bin/dokserv?id=55f13e78ab2741279d9a09a45e27b7ca&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029756709&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT meyerkennethr introductiontohamiltoniandynamicalsystemsandthenbodyproblem AT offindan introductiontohamiltoniandynamicalsystemsandthenbodyproblem AT springerinternationalpublishing introductiontohamiltoniandynamicalsystemsandthenbodyproblem |