Homogenization in time of singularly perturbed mechanical systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1998
|
| Series: | Lecture notes in mathematics
1687 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 156 S. graph. Darst. |
| ISBN: | 3540644474 |
Staff View
MARC
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| 100 | 1 | |a Bornemann, Folkmar |d 1967- |e Verfasser |0 (DE-588)120096269 |4 aut | |
| 245 | 1 | 0 | |a Homogenization in time of singularly perturbed mechanical systems |c Folkmar Bornemann |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1998 | |
| 300 | |a XII, 156 S. |b graph. Darst. | ||
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| 490 | 1 | |a Lecture notes in mathematics |v 1687 | |
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Record in the Search Index
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| adam_text | Contents
I Introduction 1
§1. The Basic Principles of Weak Convergence 5
§2. An Illustration of the Method 9
§2.1. The Model Problem 10
§2.2. Step 1: Equi Boundedness (Energy Principle) .... 10
§2.3. Step 2: The Weak Virial Theorem 12
§2.4. Step 3: Adiabatic Invariance of the Normal Actions 12
§2.5. Step 4: Identification of the Limit Mechanical System 13
§2.6. Comments on the Notions Introduced and the Result 14
II Homogenization of Natural Mechanical Systems 17
§1. The Homogenization Result 18
§1.1. Natural Mechanical Systems with a Strong Potential 18
§1.2. The Critical Submanifold 19
§1.3. Spectrally Smooth Constraining Potentials 20
§1.4. Resonance Conditions 21
§1.5. The Statement of the Homogenization Result .... 21
§1.6. Homogenization of a Specific Class of Potentials . . 22
§1.7. Remarks on Genericity 24
§1.8. A Counterexample for Flat Resonances 25
§1.9. A Counterexample for Unbounded Energy 27
§1.10. Bibliographical Remarks 28
§2. The Proof of the Homogenization Result 29
§2.1. Step 1: Equi Boundedness (Energy Principle) .... 29
§2.2. Step 2: The Weak Virial Theorem 36
§2.3. Step 3: Adiabatic Invariance of the Normal Actions 40
§2.4. Step 4: Identification of the Limit Mechanical System 49
§3. Realization of Holonomic Constraints 51
§3.1. Conditions on the Initial Values 52
§3.2. Conditions on the Constraining Potential 55
§3.3. Bibliographical Remarks 57
§4. Spectrally Nonsmooth Constraining Potentials: Takens Chaos 59
§5. Generalization: Friction and Gyroscopic Forces 66
XII Contents
III Applications 73
§1. Magnetic Traps and Mirrors 74
§2. Molecular Dynamics 76
§2.1. An Example: The Butane Molecule 78
§2.2. Relation to the Fixman Potential 83
§3. Quantum Classical Coupling: The Finite Dimensional Case 84
IV Adiabatic Results in Quantum Theory 89
§1. The Adiabatic Theorem of Quantum Mechanics 91
§1.1. The Result 91
§1.2. The Proof 93
§1.3. Extension to the Essential Spectrum 98
§1.4. Remarks on the Limit Density Operator 100
§1.5. The Example of a Two Body Hamiltonian 102
§1.6. Bibliographical Remarks 103
§2. Quantum Classical Coupling: The Infinite Dimensional Case 104
§2.1. The Singular Limit 105
§2.2. The Proof 107
§2.3. An Example 113
§2.4. Remarks on the Born Oppenheimer Approximation . 113
Appendix A: Eigenvalue Resonances of Codimension Two 115
Appendix B: Advanced Tools from Functional Analysis 117
§1. Weak* Convergence of Operator Valued Functions 117
§1.1. Trace Class Operators 117
§1.2. Lebesgue Bochner Spaces 119
§1.3. Spaces of Trace Class Operator Valued Functions . . 120
§1.4. The Trace is Not Weakly* Sequentially Continuous . 123
§1.5. Trace Class Operators on Rigged Hilbert Spaces . . 124
§2. Semi Bounded Operators and Coercive Quadratic Forms . 128
§2.1. Rollnik Potentials 129
Appendix C: Asymptotic Studies of Two Model Problems 131
§1. The Model Problem of the Introduction 131
§2. Quantum Classical Coupling (Finite Dimensional) 136
Appendix D: The Weak Virial Theorem and Localization of
Semiclassical Measures 143
Bibliography 145
List of Symbols 151
Index 153
|
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| illustrated | Illustrated |
| indexdate | 2025-02-20T06:38:14Z |
| institution | BVB |
| isbn | 3540644474 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008059416 |
| oclc_num | 246375331 |
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| physical | XII, 156 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spellingShingle | Bornemann, Folkmar 1967- Homogenization in time of singularly perturbed mechanical systems Lecture notes in mathematics Differentiable dynamical systems Homogenization (Differential equations) Homogenisierung Mathematik (DE-588)4403079-4 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Singuläre Störung (DE-588)4055100-3 gnd |
| subject_GND | (DE-588)4403079-4 (DE-588)4137931-7 (DE-588)4055100-3 |
| title | Homogenization in time of singularly perturbed mechanical systems |
| title_auth | Homogenization in time of singularly perturbed mechanical systems |
| title_exact_search | Homogenization in time of singularly perturbed mechanical systems |
| title_full | Homogenization in time of singularly perturbed mechanical systems Folkmar Bornemann |
| title_fullStr | Homogenization in time of singularly perturbed mechanical systems Folkmar Bornemann |
| title_full_unstemmed | Homogenization in time of singularly perturbed mechanical systems Folkmar Bornemann |
| title_short | Homogenization in time of singularly perturbed mechanical systems |
| title_sort | homogenization in time of singularly perturbed mechanical systems |
| topic | Differentiable dynamical systems Homogenization (Differential equations) Homogenisierung Mathematik (DE-588)4403079-4 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Singuläre Störung (DE-588)4055100-3 gnd |
| topic_facet | Differentiable dynamical systems Homogenization (Differential equations) Homogenisierung Mathematik Differenzierbares dynamisches System Singuläre Störung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008059416&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT bornemannfolkmar homogenizationintimeofsingularlyperturbedmechanicalsystems |
Indexes
THWS Schweinfurt Zentralbibliothek Lesesaal
| Call Number: |
2000 SK 950 B736 |
|---|---|
| Copy 1 | ausleihbar Available Place a Hold |