Mathematical aspects of classical and celestial mechanics:
Saved in:
| Previous Title: | Dynamical systems |
|---|---|
| Main Authors: | , , |
| Format: | Book |
| Language: | German Russian |
| Published: |
Berlin [u.a.]
Springer
1997
|
| Edition: | 2. printing of the 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 276 - 285. - Früher u.d.T.: Dynamical systems |
| Physical Description: | XIV, 291 S. graph. Darst. |
| ISBN: | 3540612246 |
Staff View
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| 245 | 1 | 0 | |a Mathematical aspects of classical and celestial mechanics |c V. I. Arnold ; V. V. Kozlov ; A. I. Neishtadt. [Transl. A. Iacob] |
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Record in the Search Index
| _version_ | 1804125558027583488 |
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| adam_text | Mathematical Aspects
of Classical and Celestial Mechanics
V.I. Arnold V.V. Kozlov A.I. Neishtadt
Translated from the Russian
by A.Iacob
Contents
Chapter 1. Basic Principles of Classical Mechanics 1
§ 1. Newtonian Mechanics 1
1.1. Space, Time, Motion 1
1.2. The Newton Laplace Principle of Determinacy 2
1.3. The Principle of Relativity 4
1.4. Basic Dynamical Quantities. Conservation Laws 6
§ 2. Lagrangian Mechanics 9
2.1. Preliminary Remarks 9
2.2. Variations and Extremals 10
2.3. Lagrange s Equations 12
2.4. Poincare s Equations 13
2.5. Constrained Motion 16
§ 3. Hamiltonian Mechanics 20
3.1. Symplectic Structures and Hamilton s Equations 20
3.2. Generating Functions 22
3.3. Symplectic Structure of the Cotangent Bundle 23
3.4. The Problem of n Point Vortices 24
3.5. The Action Functional in Phase Space 26
3.6. Integral Invariants 27
3.7. Applications to the Dynamics of Ideal Fluids 29
3.8. Principle of Stationary Isoenergetic Action 30
§ 4. Vakonomic Mechanics 31
4.1. Lagrange s Problem 32
4.2. Vakonomic Mechanics 33
VIII Contents
4.3. The Principle of Determinacy 36
4.4. Hamilton s Equations in Redundant Coordinates 37
§ 5. Hamiltonian Formalism with Constraints 38
5.1. Dirac s Problem 38
5.2. Duality 40
§ 6. Realization of Constraints 40
6.1. Various Methods of Realizing Constraints 40
6.2. Holonomic Constraints 41
6.3. Anisotropic Friction 42
6.4. Adjoining Masses 43
6.5. Adjoining Masses and Anisotropic Friction 46
6.6. Small Masses 47
Chapter 2. The w Body Problem 49
§ 1. The Two Body Problem 49
1.1. Orbits 49
1.2. Anomalies 53
1.3. Collisions and Regularization 55
1.4. Geometry of the Kepler Problem 57
§ 2. Collisions and Regularization 58
2.1. Necessary Conditions for Stability 58
2.2. Simultaneous Collisions 59
2.3. Binary Collisions 60
2.4. Singularities of Solutions in the w Body Problem 62
§ 3. Particular Solutions 64
3.1. Central Configurations 65
3.2. Homographic Solutions 65
3.3. The Amended Potential and Relative Equilibria 66
§ 4. Final Motions in the Three Body Problem 67
4.1. Classification of Final Motions According to Chazy .... 67
4.2. Symmetry of Past and Future 68
§ 5. The Restricted Three Body Problem 69
5.1. Equations of Motion. The Jacobi Integral 69
5.2. Relative Equilibria and the Hill Region 71
5.3. Hill s Problem 72
§ 6. Ergodic Theorems in Celestial Mechanics 75
6.1. Stability in the Sense of Poisson 75
6.2. Probability of Capture 76
Chapter 3. Symmetry Groups and Reduction (Lowering the Order) . . 78
§1. Symmetries and Linear First Integrals 78
1.1. E. Noether s Theorem 78
1.2. Symmetries in Nonholonomic Mechanics 82
Contents IX
1.3. Symmetries in Vakonomic Mechanics 84
1.4. Symmetries in Hamiltonian Mechanics 84
§ 2. Reduction of Systems with Symmetry 86
2.1. Lowering the Order (the Lagrangian Aspect) 86
2.2. Lowering the Order (the Hamiltonian Aspect) 91
2.3. Examples: Free Motion of a Rigid Body and the Three Body
Problem 96
§3. Relative Equilibria and Bifurcations of Invariant Manifolds . . . 101
3.1. Relative Equilibria and the Amended Potential 101
3.2. Invariant Manifolds, Regions of Possible Motions, and
Bifurcation Sets 102
3.3. The Bifurcation Set in the Planar Three Body Problem . . .104
3.4. Bifurcation Sets and Invariant Manifolds in the Motion of a
Heavy Rigid Body with a Fixed Point 105
Chapter 4. Integrable Systems and Integration Methods 107
§ 1. Brief Survey of Various Approaches to the Integrability of
Hamiltonian Systems 107
1.1. Quadratures 107
1.2. Complete Integrability 109
1.3. Normal Forms Ill
§2. Completely Integrable Systems 114
2.1. Action Angle Variables 114
2.2. Noncommutative Sets of First Integrals 118
2.3. Examples of Completely Integrable Systems 119
§3. Some Methods of Integrating Hamiltonian Systems 124
3.1. Method of Separation of Variables 124
3.2. Method of L A (Lax) Pairs 129
§4. Nonholonomic Integrable Systems 131
4.1. Differential Equations with Invariant Measure 131
4.2. Some Solved Problems of Nonholonomic Mechanics . . . .134
Chapter 5. Perturbation Theory for Integrable Systems 138
§1. Averaging of Perturbations 138
1.1. The Averaging Principle 138
1.2. Procedure for Eliminating Fast Variables in the Absence of
Resonances 142
1.3. Procedure for Eliminating Fast Variables in the Presence of
Resonances 145
1.4. Averaging in Single Frequency Systems 146
1.5. Averaging in Systems with Constant Frequencies 153
1.6. Averaging in Nonresonant Domains 155
1.7. The Effect of a Single Resonance 156
1.8. Averaging in Two Frequency Systems 161
X Contents
1.9. Averaging in Multi Frequency Systems 165
§ 2. Averaging in Hamiltonian Systems 167
2.1. Application of the Averaging Principle 167
2.2. Procedures for Eliminating Fast Variables 175
§3. The KAM Theory 182
3.1. Unperturbed Motion. Nondegeneracy Conditions 182
3.2. Invariant Tori of the Perturbed System 183
3.3. Systems with Two Degrees of Freedom 186
3.4. Diffusion of Slow Variables in Higher Dimensional Systems,
and its Exponential Estimate 189
3.5. Variants of the Theorem on Invariant Tori 191
3.6. A Variational Principle for Invariant Tori. Cantori 194
3.7. Applications of the KAM Theory 197
§ 4. Adiabatic Invariants 200
4.1. Adiabatic Invariance of the Action Variable in Single
Frequency Systems 200
4.2. Adiabatic Invariants of Multi Frequency Hamiltonian Systems 205
4.3. Procedure for Eliminating Fast Variables. Conservation Time
of Adiabatic Invariants 207
4.4. Accuracy of the Conservation of Adiabatic Invariants .... 208
4.5. Perpetual Conservation of Adiabatic Invariants 210
Chapter 6. Nonintegrable Systems 212
§ 1. Near Integrable Hamiltonian Systems 212
1.1. Poincare s Methods 213
1.2. Creation of Isolated Periodic Solutions is an Obstruction to
Integrability 215
1.3. Applications of Poincare s Method 218
§ 2. Splitting of Asymptotic Surfaces 220
2.1. Conditions for Splitting 221
2.2. Splitting of Asymptotic Surfaces is an Obstruction to
Integrability 224
2.3. Applications 227
§ 3. Quasi Random Oscillations 231
3.1. The Poincare Map 232
3.2. Symbolic Dynamics 235
3.3. Nonexistence of Analytic First Integrals 237
§ 4. Nonintegrability in the Neighborhood of an Equilibrium Position
(Siegel s Method) 238
§ 5. Branching of Solutions and Nonexistence of Single Valued First
Integrals 241
5.1. Branching of Solutions is an Obstruction to Integrability . .241
5.2. Monodromy Groups of Hamiltonian Systems with Single
Valued First Integrals 244
Contents XI
§ 6. Topological and Geometrical Obstructions to Complete
Integrability of Natural Systems with Two Degrees of Freedom 248
6.1. Topology of the Configuration Space of Integrable Systems 248
6.2. Geometrical Obstructions to Integrability 250
Chapter 7. Theory of Small Oscillations 251
§ 1. Linearization 251
§ 2. Normal Forms of Linear Oscillations 252
2.1. Normal Form of Linear Natural Lagrangian Systems .... 252
2.2. The Rayleigh Fischer Courant Theorems on the Behavior of
Characteristic Frequencies under an Increase in Rigidity and
under Imposition of Constraints 253
2.3. Normal Forms of Quadratic Hamiltonians 253
§3. Normal Forms of Hamiltonian Systems Near Equilibria .... 255
3.1. Reduction to Normal Form 255
3.2. Phase Portraits of Systems with Two Degrees of Freedom in
the Neighborhood of an Equilibrium Position under Resonance 258
3.3. Stability of Equilibria in Systems with Two Degrees of Freedom
under Resonance 264
§ 4. Normal Forms of Hamiltonian Systems Near Closed Trajectories 266
4.1. Reduction to the Equilibrium of a System with Periodic
Coefficients 266
4.2. Reduction of Systems with Periodic Coefficients to Normal
Form 267
4.3. Phase Portraits of Systems with two Degrees of Freedom Near
a Closed Trajectory under Resonance 267
§ 5. Stability of Equilibria in Conservative Fields 271
Comments on the Bibliography 274
Recommended Reading 276
Bibliography 278
Index 286
|
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| author | Arnolʹd, V. I. 1937-2010 Kozlov, Valerij V. Nejštadt, Anatolij I. |
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| edition | 2. printing of the 2. ed. |
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| id | DE-604.BV011068121 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:03:28Z |
| institution | BVB |
| isbn | 3540612246 |
| language | German Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007413585 |
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| physical | XIV, 291 S. graph. Darst. |
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| publisher | Springer |
| record_format | marc |
| spelling | Arnolʹd, V. I. 1937-2010 Verfasser (DE-588)119540878 aut Mathematical aspects of classical and celestial mechanics V. I. Arnold ; V. V. Kozlov ; A. I. Neishtadt. [Transl. A. Iacob] 2. printing of the 2. ed. Berlin [u.a.] Springer 1997 XIV, 291 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 276 - 285. - Früher u.d.T.: Dynamical systems Mécanique analytique Mécanique analytique ram Mécanique céleste Mécanique céleste ram Celestial mechanics Mechanics, Analytic Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Mechanik (DE-588)4038168-7 gnd rswk-swf Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Mechanik (DE-588)4038168-7 s Mathematische Methode (DE-588)4155620-3 s 1\p DE-604 Mathematik (DE-588)4037944-9 s 2\p DE-604 Theoretische Mechanik (DE-588)4185100-6 s Mathematische Physik (DE-588)4037952-8 s 3\p DE-604 Kozlov, Valerij V. Verfasser aut Nejštadt, Anatolij I. Verfasser aut Früher u.d.T. Dynamical systems HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007413585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Arnolʹd, V. I. 1937-2010 Kozlov, Valerij V. Nejštadt, Anatolij I. Mathematical aspects of classical and celestial mechanics Mécanique analytique Mécanique analytique ram Mécanique céleste Mécanique céleste ram Celestial mechanics Mechanics, Analytic Mathematische Physik (DE-588)4037952-8 gnd Mathematik (DE-588)4037944-9 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Mechanik (DE-588)4038168-7 gnd Mathematische Methode (DE-588)4155620-3 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4037944-9 (DE-588)4185100-6 (DE-588)4038168-7 (DE-588)4155620-3 |
| title | Mathematical aspects of classical and celestial mechanics |
| title_auth | Mathematical aspects of classical and celestial mechanics |
| title_exact_search | Mathematical aspects of classical and celestial mechanics |
| title_full | Mathematical aspects of classical and celestial mechanics V. I. Arnold ; V. V. Kozlov ; A. I. Neishtadt. [Transl. A. Iacob] |
| title_fullStr | Mathematical aspects of classical and celestial mechanics V. I. Arnold ; V. V. Kozlov ; A. I. Neishtadt. [Transl. A. Iacob] |
| title_full_unstemmed | Mathematical aspects of classical and celestial mechanics V. I. Arnold ; V. V. Kozlov ; A. I. Neishtadt. [Transl. A. Iacob] |
| title_old | Dynamical systems |
| title_short | Mathematical aspects of classical and celestial mechanics |
| title_sort | mathematical aspects of classical and celestial mechanics |
| topic | Mécanique analytique Mécanique analytique ram Mécanique céleste Mécanique céleste ram Celestial mechanics Mechanics, Analytic Mathematische Physik (DE-588)4037952-8 gnd Mathematik (DE-588)4037944-9 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Mechanik (DE-588)4038168-7 gnd Mathematische Methode (DE-588)4155620-3 gnd |
| topic_facet | Mécanique analytique Mécanique céleste Celestial mechanics Mechanics, Analytic Mathematische Physik Mathematik Theoretische Mechanik Mechanik Mathematische Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007413585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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