Classical mechanics with calculus of variations and optimal control: an intuitive introduction
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2014
|
| Series: | Student mathematical library
69 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 295 - 296 |
| Physical Description: | XX, 299 S. Ill., graph. Darst. |
| ISBN: | 9780821891384 |
Staff View
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Record in the Search Index
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| adam_text | Titel: Classical mechanics with calculus of variations and optimal control
Autor: Levi, Mark
Jahr: 2014
Contents
Series Foreword: MASS and REU at Penn State University v
Preface xiii
Chapter 1. One Degree of Freedom 1
§1. The setup 1
§2. Equations of motion 2
§3. Potential energy 6
§4. Kinetic energy 10
§5. Conservation of total energy 12
§6. The phase plane 14
§7. Lagrangian equations of motion 18
§8. The variational meaning of the Euler-Lagrange equation 19
§9. Euler-Lagrange equations — general theory 21
§10. Noether s theorem/Energy conservation 23
§11. Hamiltonian equations of motion 24
§12. The phase flow 26
§13. The divergence 28
§14. A lemrna on moving domains 31
§15. Divergence as a measure of expansion 34
viii
Contents
§16. Liouville s theorem 35
§17. The uncertainty principle of classical mechanics 36
§18. Can one hear the shape of the potential? 39
§19. A dynamics-statics equivalence 42
§20. Chapter summary 48
§21. Problems 50
Chapter 2. More Degrees of Freedom 75
§1. Newton s laws 75
§2. Center of mass 77
§3. Newton s second law for multi-particle Systems 79
§4. Angular momentum, torque 80
§5. Rotational Version of Newton s second law; conservation
of the angular momentum 81
§6. Circular motion: angular position, velocity, acceleration 84
§7. Energy and angular momentum of rotation 85
§8. The rotational - translational analogy 87
§9. Potential force fields 87
§10. Some physical remarks 90
§11. Conservation of energy 91
§12. Central force fields; conservation of angular momentum 92
§13. Kepler s problem 94
§14. Kepler s trajectories are conics: a short proof 96
§15. Motion in linear central fields 199
§16. Linear vibrations: derivation of the equations 104
§17. A nonholonomic System 195
§18. The modal decomposition of vibrations 108
§19. Lissajous figures and Chebyshev s polynomials 111
§20. Invariant 2-tori in E4 HO
§21. Rayleigh s quotient and a physical interpretation 117
§22. The Coriolis and the centrifugal forces 119
Contents
ix
§23. Miscellaneous examples 122
§24. Problems 126
Chapter 3. Rigid Body Motion 141
§1. Reference frames, angular velocity 142
§2. The tensor of inertia 143
§3. The kinetic energy 147
§4. Dynamics in the body frame 148
§5. Euler s equations of motion 150
§6. The tennis racket paradox 151
§7. Poinsot s description of free rigid body motion 152
§8. The gyroscopic effect — an intuitive explanation 154
§9. The gyroscopic torque 156
§10. Speed of precession 157
§11. The gyrocompass 159
§12. Problems 161
Chapter 4. Variational Principles of Mechanics 167
§1. The setting 167
§2. Lagrange s equations 168
§3. Examples 169
§4. Hamilton s principle 171
§5. Hamilton s principle o Euler-Lagrange equations 172
§6. Advantages of Hamilton s principle 173
§7. Maupertuis principle — some history 174
§8. Maupertuis principle on an example 175
§9. Maupertuis principle — a more general Statement 176
§10. Discussion of the Maupertuis principle 177
§11. Problems 179
Chapter 5. Classical Problems of Calculus of Variations 183
§1. Introduction and an overview 183
§2. Dido s problem — a historical note 184
X
Contents
§3. A special class of Lagrangians 185
§4. The shortest way to the smallest integral 187
§5. The brachistochrone problem 189
§6. Johann Bernoulli s Solution of the brachistochrone
problem 192
§7. Geodesics in Poincare s metric 194
§8. The soap film, or the minimal surface of revolution 196
§9. The catenary: formulating the problem 200
§10. Minimizing with constraints — Lagrange multipliers 201
§11. Catenary — the Solution 203
§12. An elementary Solution for the catenary 204
§13. Problems 205
Chapter 6. The Conditions of Legendre and Jacobi for a
Minimum 211
§1. Conjugate points 212
§2. The Legendre and the Jacobi conditions 215
§3. Quadratic functionals: the fundamental theorem 217
§4. Sufficient conditions for a minimum for a general
functional 219
§5. Necessity of the Legendre condition for a minimum 222
§6. Necessity of the Jacobi condition for a minimum 223
§7. Some intuition on positivity of functionals 226
§8. Problems 229
Chapter 7. Optimal Control 233
§1. Formulation of the problem 233
§2. The Maximum Principle 235
§3. A geometrical explanation of the Maximum Principle 237
§4. Example 1: a smooth landing 244
§5. Example 2: stopping a harmonic oscillator 246
§6. Huygens s principle vs. Maximum Principle 250
Contents
xi
§7. Background on linearized and adjoint equations 252
§8. Problems 254
Chapter 8. Heuristic Foundations of Hamiltonian Mechanics 257
§1. Some fundamental questions 257
§2. The main idea 258
§3. The Legendre transform, the Hamiltonian, the
momentum 263
§4. Properties of the Legendre transform 264
§5. The Hamilton-Jacobi equation 266
§6. Noether s theorem 267
§7. Conservation of energy 270
§8. Poincare s integral invariants 271
§9. The generating function 273
§10. Hamilton s equations 273
§11. Hamiltonian mechanics as the spring theory 274
§12. The optical-mechanical analogy 280
§13. Hamilton-Jacobi equation leading to the Schrödinger
equation 284
§14. Examples and Problems 287
Bibliography 295
Index 297
|
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| illustrated | Illustrated |
| indexdate | 2024-07-10T00:59:52Z |
| institution | BVB |
| isbn | 9780821891384 |
| language | English |
| lccn | 013030550 |
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| publisher | American Math. Soc. |
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| series | Student mathematical library |
| series2 | Student mathematical library |
| spelling | Levi, Mark 1951- Verfasser (DE-588)172238099 aut Classical mechanics with calculus of variations and optimal control an intuitive introduction Mark Levi Providence, RI American Math. Soc. 2014 XX, 299 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Student mathematical library 69 Literaturverz. S. 295 - 296 Ordinary differential equations msc Dynamical systems and ergodic theory msc Calculus of variations and optimal control; optimization msc Mechanics of particles and systems msc General ... General and miscellaneous specific topics ... Problem books msc Mechanics Textbooks Hamiltonian systems Textbooks Calculus of variations Textbooks Control theory Textbooks Ordinary differential equations Dynamical systems and ergodic theory Calculus of variations and optimal control; optimization Mechanics of particles and systems General ... General and miscellaneous specific topics ... Problem books Kontrolltheorie (DE-588)4032317-1 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Mechanik (DE-588)4038168-7 gnd rswk-swf Mechanik (DE-588)4038168-7 s Hamiltonsches System (DE-588)4139943-2 s Variationsrechnung (DE-588)4062355-5 s Kontrolltheorie (DE-588)4032317-1 s DE-604 Student mathematical library 69 (DE-604)BV013184751 69 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027016167&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Levi, Mark 1951- Classical mechanics with calculus of variations and optimal control an intuitive introduction Student mathematical library Ordinary differential equations msc Dynamical systems and ergodic theory msc Calculus of variations and optimal control; optimization msc Mechanics of particles and systems msc General ... General and miscellaneous specific topics ... Problem books msc Mechanics Textbooks Hamiltonian systems Textbooks Calculus of variations Textbooks Control theory Textbooks Ordinary differential equations Dynamical systems and ergodic theory Calculus of variations and optimal control; optimization Mechanics of particles and systems General ... General and miscellaneous specific topics ... Problem books Kontrolltheorie (DE-588)4032317-1 gnd Hamiltonsches System (DE-588)4139943-2 gnd Variationsrechnung (DE-588)4062355-5 gnd Mechanik (DE-588)4038168-7 gnd |
| subject_GND | (DE-588)4032317-1 (DE-588)4139943-2 (DE-588)4062355-5 (DE-588)4038168-7 |
| title | Classical mechanics with calculus of variations and optimal control an intuitive introduction |
| title_auth | Classical mechanics with calculus of variations and optimal control an intuitive introduction |
| title_exact_search | Classical mechanics with calculus of variations and optimal control an intuitive introduction |
| title_full | Classical mechanics with calculus of variations and optimal control an intuitive introduction Mark Levi |
| title_fullStr | Classical mechanics with calculus of variations and optimal control an intuitive introduction Mark Levi |
| title_full_unstemmed | Classical mechanics with calculus of variations and optimal control an intuitive introduction Mark Levi |
| title_short | Classical mechanics with calculus of variations and optimal control |
| title_sort | classical mechanics with calculus of variations and optimal control an intuitive introduction |
| title_sub | an intuitive introduction |
| topic | Ordinary differential equations msc Dynamical systems and ergodic theory msc Calculus of variations and optimal control; optimization msc Mechanics of particles and systems msc General ... General and miscellaneous specific topics ... Problem books msc Mechanics Textbooks Hamiltonian systems Textbooks Calculus of variations Textbooks Control theory Textbooks Ordinary differential equations Dynamical systems and ergodic theory Calculus of variations and optimal control; optimization Mechanics of particles and systems General ... General and miscellaneous specific topics ... Problem books Kontrolltheorie (DE-588)4032317-1 gnd Hamiltonsches System (DE-588)4139943-2 gnd Variationsrechnung (DE-588)4062355-5 gnd Mechanik (DE-588)4038168-7 gnd |
| topic_facet | Ordinary differential equations Dynamical systems and ergodic theory Calculus of variations and optimal control; optimization Mechanics of particles and systems General ... General and miscellaneous specific topics ... Problem books Mechanics Textbooks Hamiltonian systems Textbooks Calculus of variations Textbooks Control theory Textbooks Kontrolltheorie Hamiltonsches System Variationsrechnung Mechanik |
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