Introduction to the calculus of variations and its applications:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York u.a.
Chapman & Hall
1995
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 638 S. graph. Darst. |
| ISBN: | 0412051419 |
Internformat
MARC
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| 100 | 1 | |a Wan, Frederic Y. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to the calculus of variations and its applications |c Frederick Y. M. Wan |
| 264 | 1 | |a New York u.a. |b Chapman & Hall |c 1995 | |
| 300 | |a XVIII, 638 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Calculus of variations | |
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Datensatz im Suchindex
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| adam_text | Contents
Page
Preface xiii
1. The Basic Problem
1. Introduction 1
2. Some Examples 3
3. The Euler Differential Equation 9
4. Integration of the Euler Differential Equation 14
5. The Brachistochrone Problem 22
6. Piecewise Smooth Extremals 26
7. Exercises 28
2. Piecewise Smooth Extremals
1. Piecewise Smooth Solution for the Basic Problem 34
2. The Euler Lagrange Equation 37
3. Several Unknowns 39
4. Parametric Form 42
5. Erdmann s Corner Conditions 45
6. The Ultra Differentiated Form 48
7. Minimal Surface of Revolution 49
8. Maximum Rocket Height 53
9. Exercises 55
3. Modifications of the Basic Problem
1. The Variational Notation 58
2. Euler Boundary Conditions 61
3. Free Boundary Problems 65
vii
viii Contents
4. Free and Constrained End Points 68
5. Higher Derivatives 71
6. Other End Conditions 76
7. Exercises 79
4. A Weak Minimum
1. The Legendre Condition 85
2. Jacobi s Test 89
3. Conjugate Points 92
4. Sufficiency 96
5. Several Unknowns 99
6. Convex Integrand 104
7. Global Minimum 107
8. Exercises 109
5. A Strong Minimum
1. A Weak Minimum May Not Be the True Minimum 113
2. The Weierstrass Excess Function 115
3. The Figurative 117
4. Fields of Extremals 120
5. Sufficiency 124
6. An Illustrative Example 128
7. Hilbert s Integral 130
8. Several Unknowns 133
9. Exercises 135
Appendix 137
6. The Hamiltonian
1. The Legendre Transformation and Hamiltonian Systems 139
2. Hamilton s Principle 142
3. Canonical Transformations 145
4. The Hamilton Jacobi Equation 148
5. Solutions of the Hamilton Jacobi Equation 152
6. The Method of Additive Separation 155
7. Hamilton s Principal Function 160
8. Exercises 164
7. Lagrangian Mechanics
1. Generalized Coordinates 167
2. Coordinate Transformations 170
Contents ix
3. Holonomic Constraints 176
4. Poisson Brackets 180
5. Variationally Invariant Lagrangians 182
6. Noether s Theorem 184
7. Generators for Variationally Invariant Lagrangians 188
8. Relativistic Mechanics 191
9. Exercises 194
8. Direct Methods
1. The Rayleigh Ritz Method 198
2. Completeness and Minimizing Sequence 201
3. A Weighted Least Squares Approximation 205
4. Inhomogeneous End Conditions 207
5. Piecewise Linear Finite Elements 212
6. The Finite Element Method 216
7. Duality 218
8. The Inverse Problem 223
9. Weak Solutions 228
10. Exercises 230
9. Dynamic Programming
1. The Shortest Route Problem 233
2. Backward Recursion 238
3. The Knapsack Problem 242
4. Forward Recursion 245
5. Intermediate Knapsack Capacities 248
6. Vector and Continuous State Variables 250
7. The Variational Problem 257
8. Exercises 261
10. Isoperimetric Constraints
1. The Shape of the Hanging Chain 266
2. Normal Isoperimetric Problems and a Duality 270
3. Eigenvalue Problems and Mechanical Vibration 273
4. Variational Formulation of Sturm Liouville Problems 277
5. The Rayleigh Quotient 280
6. Higher Eigenvalues 283
7. Mixed End Conditions 285
8. Optimal Harvesting of a Uniform Forest 286
9. Exercises 289
Appendix 293
x Contents
11. Pointwise Constraints on Extremals
1. Pointwise Equality Constraints 309
2. The Multiplier Rule for Equality Constraints 313
3. Inequality Constraints on the Unknowns 317
4. Binding Inequality Constraints 320
5. Brachistochrone with Limited Descent 323
6. Inequality Constraints on an End Point 326
7. Land Use in a Long and Narrow City 328
8. Exercises 333
12. Nonholonomic Constraints
1. Equality Constraints Involving Derivatives 338
2. The Multiplier Rule 342
3. Brachistochrone in a Resisting Medium 345
4. Inequality Constraints 350
5. Singular Solutions 355
6. The Most Rapid Approach 357
7. The Hamilton Jacobi Inequality 361
8. Blocked Harvest of a Uniform Forest 365
9. Exercises 369
13. Optimal Control with Linear Dynamics
1. Optimal Control 372
2. Statement of the Problem 375
3. Controllability of Linear Autonomous Systems 377
4. Nonautonomous Linear Systems 382
5. Controllability with Constrained Controls 385
6. An Inventory Control Model 388
7. A Wheat Trading Problem 391
8. The Hamiltonian 394
9. The Linear Time Optimal Problem 397
10. Exercises 401
14. Optimal Control with General Lagrangians
1. The Maximum Principle 405
2. Controllability of Nonlinear Systems 410
3. Sustained Consumption with a Finite Resource Deposit 413
4. The Linear Quadratic Problem and Feedback Control 417
5. A Sufficient Condition for Optimality 421
Contents xi
6. Household Optimum and Locational Equilibrium 424
7. The Second Best Residential Land Allocation 428
8. Perturbation Solution 431
9. Inequality Constraints 436
10. Optimality Under Constraints 439
11. Methods of the Calculus of Variations 443
12. Exercises 447
Appendix 449
15. Higher Dimensions
1. The Plateau Problem 451
2. Euler Differential Equation and Boundary Conditions 453
3. Sufficient Conditions 457
4. Dirichlet s Problem on a Unit Disk 459
5. Several Unknowns 461
6. Maxwell s Equations 463
7. Higher Derivatives 465
8. Finite Elements in Two Dimensions 468
9. Torsion of Elastic Bars 475
10. Pointwise Equality Constraints 483
11. Isoperimetric Constraints 486
12. Exercises 488
16. Linear Theory of Elasticity
1. Continuum Mechanics and Elasticity Theory 492
2. Components of Displacement and Strain 493
3. Stress Fields and Equilibrium 498
4. Elasticity and Isotropy 503
5. Navier s Reduction 509
6. Minimum Potential Energy 511
7. Reissner s Variational Principle 513
8. Minimum Complementary Energy 515
9. Semi direct Method 518
10. Saint Venant Torsion 522
11. Exercises 526
17. Plate Theory
1. The Elastostatics of Flat Plates 533
2. The Germain Kirchhoff Thin Plate Theory 536
3. The Kirchhoff Contracted Stress Boundary Conditions 539
xii Contents
4. A Semi direct Method of Solution 542
5. Minimum Complementary Energy 544
6. Reduction of Reissner s Plate Equations 549
7. A Variational Principle for Stresses and Displacements 552
8. Twisting of a Rectangular Plate 554
9. A Finite Deflection Plate Theory 557
10. The von Karman Plate Equations 562
11. Finite Twisting and Bending of Rectangular Plates 564
12. Exercises 568
18. Fluid Mechanics
1. Mass and Entropy 572
2. A Lagrangian Variational Principle for Ideal Fluids 574
3. Ideal Fluid Motion Not Always Irrotational 577
4. An Eulerian Variational Principle for Ideal Fluids 578
5. Incompressible Fluids 580
6. A Surface Wave Problem 583
7. Slow Dispersion of Wave Trains 588
8. Creeping Motion of an Incompressible Fluid 593
9. Oseen s Approximation 596
10. Exercises 597
Appendix. Approximate Methods for Euler s Differential Equation
1. Two Point Boundary Value Problems 599
2. Numerical Solution for Initial Value Problems 601
3. Linear Boundary Value Problems 603
4. The Shooting Method 604
5. Finite Difference Analogue 607
6. Accuracy of the Finite Difference Solution 611
7. Fixed Point Iteration 612
8. Newton s Iteration 616
9. Exercises 617
Bibliography 621
Index 627
|
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| indexdate | 2024-07-09T17:50:48Z |
| institution | BVB |
| isbn | 0412051419 |
| language | English |
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| spelling | Wan, Frederic Y. Verfasser aut Introduction to the calculus of variations and its applications Frederick Y. M. Wan New York u.a. Chapman & Hall 1995 XVIII, 638 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Calculus of variations Berechnung (DE-588)4120997-7 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Schwingung (DE-588)4053999-4 gnd rswk-swf Schwingung (DE-588)4053999-4 s Berechnung (DE-588)4120997-7 s DE-604 Variationsrechnung (DE-588)4062355-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006882220&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wan, Frederic Y. Introduction to the calculus of variations and its applications Calculus of variations Berechnung (DE-588)4120997-7 gnd Variationsrechnung (DE-588)4062355-5 gnd Schwingung (DE-588)4053999-4 gnd |
| subject_GND | (DE-588)4120997-7 (DE-588)4062355-5 (DE-588)4053999-4 |
| title | Introduction to the calculus of variations and its applications |
| title_auth | Introduction to the calculus of variations and its applications |
| title_exact_search | Introduction to the calculus of variations and its applications |
| title_full | Introduction to the calculus of variations and its applications Frederick Y. M. Wan |
| title_fullStr | Introduction to the calculus of variations and its applications Frederick Y. M. Wan |
| title_full_unstemmed | Introduction to the calculus of variations and its applications Frederick Y. M. Wan |
| title_short | Introduction to the calculus of variations and its applications |
| title_sort | introduction to the calculus of variations and its applications |
| topic | Calculus of variations Berechnung (DE-588)4120997-7 gnd Variationsrechnung (DE-588)4062355-5 gnd Schwingung (DE-588)4053999-4 gnd |
| topic_facet | Calculus of variations Berechnung Variationsrechnung Schwingung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006882220&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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