INTRODUCTION TO THE CALCULUS OF VARIATIONS AND ITS APPLICATIONS Frederick Y.M. Wan University of California, Irvine CHAPMAN & HALL I(J)P An International Thomson Publishing Company New York Albany Bonn Boston Cincinnati Detroit London Madrid Melbourne Mexico City Pacific Grove Paris San Francisco Singapore Tokyo Toronto Washington
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 Jhe 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 Appendix. 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 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