A first course in the calculus of variations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2014]
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| Series: | Student mathematical library
volume 72 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | x, 298 Seiten Illustrationen, Diagramme |
| ISBN: | 9781470414955 |
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| 245 | 1 | 0 | |a A first course in the calculus of variations |c Mark Kot |
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Record in the Search Index
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| adam_text | Titel: A first course in the calculus of variations
Autor: Kot, Mark
Jahr: 2014
Contents Preface ix Chapter 1. Introduction 1 §1.1. The brachistochrone 1 §1.2. The terrestrial brachistochrone 5 §1.3. Geodesics 7 §1.4. Minimal surfaces 14 §1.5. Recommended reading 18 §1-6. Exercises 20 Chapter 2. The First Variation 27 §2.1. The simplest problem 27 §2.2. Euler’s approach 28 §2.3. Lagrange’s approach 32 COO to Recommended reading 45 §2.5. Exercises 46 Chapter 3. Cases and Examples 47 §3.1. Special cases 47 §3.2. Case study: Minimal surface of revolution 52 §3.3. Case study: The brachistochrone 59 v
VI Contents §3.4. Geodesics 62 §3.5. Recommended reading 63 §3.6. Exercises 64 Chapter 4. Basic Generalizations 69 §4.1. Higher-order derivatives 69 §4.2. Case study: The cantilever beam 71 §4.3. Multiple unknown functions 77 §4.4. Lagrangian mechanics 79 §4.5. Case study: The spherical pendulum 85 §4.6. Hamiltonian mechanics 88 §4.7. Ray optics 93 §4.8. Double integrals 96 §4.9. Recommended reading 101 §4.10. Exercises 101 Chapter 5. Constraints 107 §5.1. Types of constraints 107 §5.2. Lagrange multipliers 114 §5.3. Isoperimetric constraints 117 §5.4. Case study: Queen Dido’s problem 120 §5.5. Case study: Euler’s elastica 123 §5.6. Holonomic constraints 129 §5.7. Case study: A sliding rod 133 §5.8. Recommended reading 136 §5.9. Exercises 136 Chapter 6. The Second Variation 139 §6.1. Introduction 139 §6.2. Legendre’s condition 142 §6.3. Jacobi’s condition 147 §6.4. Case study: The catenoid revisited 160
Contents vii §6.5. Recommended reading 167 §6.6. Exercises 168 Chapter 7. Review and Preview 171 §7.1. Introduction 171 §7.2. Necessary conditions 172 §7.3. Sufficient conditions 173 §7.4. Two dependent variables 173 §7.5. History and preview 175 §7.6. Recommended reading 176 Chapter 8. The Homogeneous Problem 177 §8.1. Integrals in parametric form 177 §8.2. Euler-Lagrange equations 180 §8.3. The Weierstrass equation 182 §8.4. Case study: The parametric Queen Dido problem 186 §8.5. Recommended reading 189 §8.6. Exercises 190 Chapter 9. Variable-Endpoint Conditions 191 §9.1. Natural boundary conditions 191 §9.2. Transversality conditions 197 §9.3. Focal points 206 §9.4. Case study: Neile’s parabola 206 §9.5. Recommended reading 211 §9.6. Exercises 212 Chapter 10. Broken Extremals 215 §10.1. The Weierstrass-Erdmann corner conditions 215 §10.2. Carathéodory’s indicatrix 224 §10.3. Recommended reading 231 §10.4. Exercises 232
Contents viii Chapter 11. Strong Variations 235 §11.1. Troubles with weak variations 235 §11.2. Weierstrass’s condition 242 §11.3. Case study: Newton’s problem 247 §11.4. Recommended reading 254 §11.5. Exercises 254 Chapter 12. Sufficient Conditions 255 §12.1. Introduction 255 §12.2. Fields of extremals 256 §12.3. Hilbert’s invariant integral 260 §12.4. Weierstrass’s E-function revisited 262 §12.5. The royal road 267 §12.6. Recommended reading 273 §12.7. Exercises 274 Bibliography 277 Index 295
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| institution | BVB |
| isbn | 9781470414955 |
| language | English |
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| spelling | Kot, Mark 1956- Verfasser (DE-588)142984094 aut A first course in the calculus of variations Mark Kot Providence, Rhode Island American Mathematical Society [2014] © 2014 x, 298 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Student mathematical library volume 72 Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1961-5 Student mathematical library volume 72 (DE-604)BV013184751 72 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027937163&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kot, Mark 1956- A first course in the calculus of variations Student mathematical library Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4062355-5 |
| title | A first course in the calculus of variations |
| title_auth | A first course in the calculus of variations |
| title_exact_search | A first course in the calculus of variations |
| title_full | A first course in the calculus of variations Mark Kot |
| title_fullStr | A first course in the calculus of variations Mark Kot |
| title_full_unstemmed | A first course in the calculus of variations Mark Kot |
| title_short | A first course in the calculus of variations |
| title_sort | a first course in the calculus of variations |
| topic | Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Variationsrechnung |
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